What is the explanation (according to the Jewish Bible, Talmud, etc.) for the destruction of the First Temple?
Idolatry, immorality and murder. See Babylonian Talmud, Yoma 9b:
מקדש ראשון מפני מה חרב מפני ג' דברים שהיו בו ע"ז וגלוי עריות ושפיכות דמים
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.