What is the explanation (according to the Jewish Bible, Talmud, etc.) for the destruction of the First Temple?
The Jews distanced themselves from God, who therefore caused the symbol and medium of their closeness to him to be destroyed. More specifically, the g'mara (according to Rashi there) says, they did not view the Tora as important.
The most famous reason is committing the 3 cardinal sins, but other reasons are given also. The Bible gives some reasons in a couple of places, and the Talmud expounds on them. The Pentateuch has many warnings if the Jews are bad, they will be exiled (See the weekly reading next week). One of the sins emphasized is the failure to observe the Sabbatical year. Therefore it is said that the Jews were exiled for 70 years for failing to observe 70 sabbaticals.
A verse in the prophets mentions how they abandoned the Torah. Chazal say that this refers to the failure to recite the blessing over the Torah. Many commentators explain what this can mean (i.e what aspect of Torah study did they fail at).
Whatever this failure was, their abandonment of the Torah is likely connected to their eventual committing of the 3 cardinal sins.
I'd argue that the first temple was really in the city of Shiloh, but this aside, the Rambam wrote:
This is why our kingdom was lost and our Temple was destroyed and why we were brought to this; for our fathers sinned and are no more because they found many books dealing with these themes of the star gazers, these things being the root of idolatry, as we have made clear in Laws Concerning Idolatry. They erred and were drawn after them, imagining them to be glorious science and to be of great utility. They did not busy themselves with the art of war or with the conquest of lands, but imagined that those studies would help them. Therefore the prophets called them “fools and dolts.”