In ancient times, Jewish faith and practice were different from what they are today. Most people know that. However, they usually don't know the details and don't think about the implications.
In the beginnings of our faith, just as for most theistic faiths, we thought of God in anthropomorphic terms. He had a physical body. He was finite, more powerful than humans but not omnipotent, and He lived in the universe but had not created it out of nothing:
"Both Christians and Jews, each in their own way, have even accepted God’s physical attributes without much care ... Pre-philosophical Jews and Christians accepted both psychic and somatic anthropomorphism as a root principle of their faith." 
That primitive idea of God makes our central Jewish beliefs relatively straightforward -- at least on the level of meaning, if not of historical and archaeological evidence.
"God gave the Torah to Moses at Sinai" meant that a finite, visible being with a physical body, who lives in the universe but did not create it, dictated a book to a man. It's quite conceivable. "God chose the Jewish people" meant that the same being made a covenant with the ancient Israelites, following the structure used by the Hittite emperor for suzerain treaties with vassal states. 
However, as our concept of God evolved, it caused problems for our traditional beliefs.
Starting as an anthropomorphic national god similar to other national gods of the Ancient Near East, He was reimagined as the Creator ex nihilo of the universe. He was infinite, transcendent, and utterly "other." That had some philosophical merit, but it made our traditional beliefs as incomprehensible as the God to whom they referred, who had:
"... been reduced—or elevated, according to one’s own personal taste—to an impersonal principle: Omniscient, Omnipotent, All-Good, Infinite, and so on ... Philosophy has lost its radical doubt (God is still affirmed as a person), while myth has lost its fire (God is not much of a person)." 
Making God transcendent and incomprehensible makes belief that "God gave the Torah to Moses at Sinai" and other theistic beliefs into nonsense, because their subject is unknown and unknowable. According to Maimonides, the verb is also unknowable:
"His essential attributes ... must not be like the attributes of other beings ... Similarly the terms 'knowledge,' 'power,' 'will,' and 'life,' as applied to Him, may He be exalted, and to all those possessing knowledge, power, will, and life, are purely equivocal, so that their meaning when they are predicated of Him is in no way like their meaning in other applications." 
Therefore, we end up with "Blank blank the Torah to Moses at Sinai." The belief no longer has any obvious logical meaning.
The belief is also immune to empirical testing. The parts that have ordinary meaning ("... the Torah to Moses at Sinai") imply that there should be historical and archaeological evidence of corresponding events such as the Exodus. Such evidence either does not exist, is not adequate, or is contradicted by the evidence we have. 
So we have a belief that, considered as a whole, seems meaningless. It doesn't assert any facts we can understand. The parts that have ordinary meaning are believed in spite of absent or contrary empirical evidence. Whatever it is, it's not a normal belief like "Joe gave the book to Sarah." But people say that their beliefs about God have meaning for them. They're even willing to die for those beliefs. What can the meaning be?
There are at least three answers. The first is "theological incorrectness." In ordinary life, we think of God anthropomorphically even if we know better. We do it because it's efficient. It helps us to solve moral problems without getting lost in philosophical complications about God's true nature. 
The second answer is that beliefs can be meaningful in more than one way. Usually, it's by pointing to other beliefs: the belief that "Joe has brown hair" points to the belief that "Joe has hair." However, beliefs can also point to behavior. If we believe that God commanded us to keep the Sabbath, it points to our doing certain things that are morally and socially helpful. People feel that rejecting the belief would reject all of its positive consequences as well. Understandably, they don't want to reject either.
The third answer is that beliefs about God do point logically to other beliefs -- but only to beliefs about God and related concepts. That kind of meaning is limited (in our minds) and hard to explain, but it does exist.
Beliefs about God are meaningful, though sometimes not in the way we think. They're meaningful because they help us in our decisions, our lives, and our communities. For most of us, that's quite enough.
Brettler, M. et al, editors (2014), The Jewish Study Bible, second edition. Oxford University Press, Oxford, UK. Kindle edition.
Kugel, J. (2007), How to Read the Bible. Free Press, New York.
Muffs, Y. (2005), The Personhood of God. Jewish Lights Publishing, Woodstock, VT. Kindle edition.
Pines, S. translator (1963), The Guide of the Perplexed, Volume 1. University of Chicago Press, Chicago.
Slone, D.J. (2004), Theological Incorrectness: Why Religious People Believe What They Shouldn't. Oxford University Press, Oxford, UK. Kindle edition.
1. Muffs, Y. (2005), loc. 182.
2. Kugel, J. (2007), p. 243.
3. Muffs, Y. (2005), loc. 195.
4. Pines, S. (1963), p. 131.
5. Brettler, M. et al (2014), loc. 5231.
6. Slone, D.J. (2004), loc. 595ff.