There are many reasons why I turned to Judaism. I'm not sure if I mentioned this elsewhere or not, but I am a convert to Judaism. My reasons for joining the covenant are a mix of personal revelations and just logical analysis. The personal revelation part is what pushed me to investigate further, but it's what I found that actually kept me on track. So out of respect for your request, the things I found after analyzing Judaism are what I will focus on. It basically boils down to four things: logic, historicity, the history of the Jewish people, and prophecy. Since the logic of it all is interconnected with the others, I'll discuss it as I go along.
In a former life, I was a scientist of sorts. The things I learned in the lab about the nature of scientific theory and the study of the universe have stuck with me, and I naturally apply them to everything I investigate. Judaism was--and still is--no different. Being this way, I thought it important to determine the historical accuracy of as much of the Tanakh as possible if it claimed to be a literal account of the history of the Jewish people.
The first thing I did was determine which parts of the Tanakh are claimed to be literal historical accounts, and which parts--if any--are entirely or partially metaphorical. One of the first things I found was that much of the Tanakh is either metaphorical or intentionally non-chronological, particularly the Torah--it simply is not meant to be a history book.
However, despite this, it is taken as--for the most part--a history of the Jewish people. While there is debate as to whether or not the story of Creation and the Garden is supposed to be taken literally (many opinions even back into Second Temple times say that it is a metaphor), it is not debated that Noach onward happened (important to note is that the ages are NOT necessarily literal, as well as other aspects of those stories--what is true is that they actually happened). And so, finding some corroboration of this was important to me.
What I found is that while there is not direct evidence of much of the Torah, there have been some interesting things found. For instance, they have statues of a ram caught in a thicket from about the time the Torah says Abraham lived--from the area he lived. For those unfamiliar with Abraham, one of the most important things he did was show complete faith in G-d by agreeing to offer his only son Isaac as a burnt offering to G-d. When he was about to take the knife to his son's throat, an angel stopped him and revealed a ram caught in a thicket, which took the place of his son. Considering Judaism places the last words of the Torah being written in 1273 BCE, and Abraham lived nearly a thousand years before that, the statues indicate that the event may have actually occurred. In addition, they have found documents (tablets) with the names of Abraham, Sarah, Isaac, and Terah (Abraham's father) written on them in those areas they were said to have lived. Again, not direct proof of their existence, but certainly suggestive.
To briefly cover the rest, since I don't want to take up more time, they have evidence of a large deluge (and while the Torah says the flood destroyed the entire world, the Midrash clarifies this by outlining exactly where the flood took place--which was not literally the entire planet), of the Tower of Babel (entries by a Greek historian who visited its location and wrote detailed descriptions), of Semitic slaves in Egypt (paintings, papyrus scrolls discussing slaves building the cities quoted and not meeting their quota of bricks--mentioned in the Torah), and evidence of an exodus by different groups of people from Egypt (and the Torah is clear that it was not only Jews who left Egypt in the Exodus).
As far as the Tanakh goes, one needs only to go to Israel to see the history of the Jewish people matches that of the Tanakh. The Temples, the Kings, the cities, etc. There is much archeological evidence to prove what is written is accurate. It's easy to find, so I won't bother mentioning it all here. However, what I will mention is the historical consistency of the Torah. Every time they find a Torah scroll, it is exactly the same as the Torah scrolls we have today--word for word, down to the script used to write each letter. From the copies of the Torah and Isaiah from over 2,000 years ago to Torah scrolls from the 7th century BCE, the consistency is perfect (in fact, you can read the Isaiah scroll from 2,000 years ago online and compare it to the scrolls/books from today--they match perfectly).
In fact, of all the Torah scrolls around the world for the last 3,000 years, there are only 9 letter differences. These 9 are all in Yemenite Torah scrolls because they were not part of the world checking system for hundreds of years. There are 304,805 letters in the Torah, and only 9 letter differences in one small community; and more importantly, the differences are spelling differences, such as the difference between color and colour--they are not word or meaning changes. Compare this with the Xtian bible, which is 1,700 years younger than the Torah, and the Xtians have not experienced anything remotely close to the 2,000 years of exile and persecution that the Jewish people have faced: in the New Testament, there are some 200,000 variants in the existing manuscripts, representing 400 variants that cause doubt about textual meaning, 50 of which are of great significance.
The fact that the same text existed not only 2,700 years ago, but more importantly, a mere 500 years after the last words of the original Torah written by Moses indicates that it is entirely plausible--completely ignoring the religious aspect--that the original Torah was written at that time--1313-1273 BCE. That leads to the next part. If the Torah was written at the time the Jewish people were in the desert, it is completely illogical to assume that someone could write a book describing events that never happened, and that everyone would somehow accept it as true. It simply makes no sense.
I've touched on the history of the Jewish people and its matching up with the Tanakh; however, I want to discuss it more in relation to prophecy. There are several specific prophecies in the Torah and Tanakh regarding the Jewish people. It would be a huge blow for the Truth of the Torah if they didn't come true. There are several, and I have discussed many of them before. Without delving too deeply into it once again, there are several prophecies regarding the Jewish people that have come true, including ones that either go against common sense or are extremely detailed. These include prophecies regarding the population size of the Jewish people, world-wide anti-Semitism, the exiles, the return to Israel, and others. Two of the most telling, however, are the ones about the Land of Israel itself and the Temples.
The prophecy regarding the Land of Israel links the prosperity of the land with the Jews. It says that when the Jews are in Israel, it will flourish; but, it will be a wasteland when the Jews are not in Israel. Before the Roman exile, Israel was very agricultural. Shortly after the exile began, the land turned to a rocky, barren desert despite the Arabs' attempts to cultivate it. In fact, when Mark Twain visited in the 1800s, he expressed wonder that anyone could live there. There were small groups of Jews living in the land who were able to make do, but the majority of Jews lived elsewhere due to the exile. When the Jews finally returned in droves to Israel not too long ago, agriculture picked up, and it is now one of the biggest--if not the biggest--industries there. There is no logical reason for this, except that it is a direct prophecy that come exactly true.
The prophecy regarding the Temples says that 70 years after the destruction of the First Temple, a Persian King named Cyrus would rebuild the Temple, and 484 years after that, a foreign invader would destroy the Second Temple and kill the high priest. So, what happened? Exactly that. 70 years after the destruction of the First Temple, King Cyrus from Persia gave the Jews permission to rebuilt the Temple. 484 years later, Titus came down from Rome and destroyed the Second Temple and killed the High Priest.
And so how does this all fit together? It's simply logic. There is historical evidence of Biblical events happening. From Joshua on, the history is entirely accurate. The Torah scrolls from the 7th century BCE are exactly the same as the ones we have today. Prophecies in the Torah have come true exactly. Put that together and what you get is a large amount of evidence that there were Jews in the desert at the time the Bible purports who believed in the words of the Torah being true. That many people could simply not be convinced that a series of events based on them happened if they didn't--it's just not possible.
As far as the more spiritual stuff goes, it also boils down to logic. Judaism is the only religion whose Revelation of G-d was not given to a single person, but rather 3 million people at once. It is also the only all-inclusive religion regarding both the afterlife and the World to Come. In Jewish faith, everyone goes to Heaven regardless of religion, and non-Jews who follow the 7 Noachide Laws are guaranteed a place in the World to Come. Focusing just on Heaven, why would G-d create a world and people for millions of years only to send them to Hell because they were born before Jebus? That makes no sense. In fact, the very concept of Hell makes no sense--which is why Judaism has no concept of it. If G-d created the universe and everything in the spiritual plane out of the kindness of His "heart", why would He create a place for us to burn and suffer forever because we sinned? If He is forgiving and knows we are prone to sin, it makes no sense that He would send us to Hell.
In addition, we have miracles. While the miracles of the Bible--specifically the Exodus--can be questioned because of how long ago they occurred, modern miracles are harder to explain away. The Jewish people experienced the most open miracles in history since the Exodus during the Six Day War in 1967. There is story after story about how two Jews were spared from dozens of Egyptian soldiers because the Egyptians became paralyzed when they went to fire on the Jews, or how the IAF was able to completely destroy the Egyptian and Jordanian air fields because warning messages were messed up or never delivered, or trucks filled with missiles and explosives were hit with mortars that simply didn't go off when they should have. The covenant between G-d and the Jewish people is that even in exile, He will ensure that we are never destroyed. Our enemies have been trying to destroy us for 2,000 years now, and have been trying to destroy the State of Israel for over 60 years now without success. As many a non-Jew (and Jew) has said, the fact that we're still around is proof that G-d exists.