Why did God allow the destruction of His temple?

Answer 1

Since the question doesn't specify, we'll discuss both the destruction of the First Temple and the Second Temple. In each case, we'll give 1) a spiritual reason, and 2) a political ("current events-type") reason.
The First Destruction (some 2500 years ago):
1) The general spiritual reason was that God found the Jewish people to be below the spiritual level that was a requisite for having His presence among them. The prophets had warned them (Jeremiah 7:25) but were not sufficiently heeded (2 Chronicles 36:16). Once God's presence no longer felt fully welcome in the Holy Temple, its destruction was just a matter of time (see Talmud, Rosh Hashanah 31a, and Eichah Rabbah 1:43).
A more specific spiritual reason was the sins of Menasheh, king of Judah, as stated in 2 Kings 21:11-13 and 24:3.
2) The political reason (the specific temporal circumstance) was the fact that King Yehoiakim, after having been obedient to Nevuchadnezzar (king of Babylonia), became insubordinate (2 Kings 24:1), and Nevuchadnezzar responded by conquering the land of Judah and destroying the First Temple.
The Second Destruction (in 68 CE):
1) The spiritual reason was that the Jewish people were plagued by an internal enemy, the Sadducees, who had been dragging the spiritual level of the people downwards for many decades. These were men of power who (along the lines of the earlier Hellenizers) were less interested in Torah than in pleasure, politics, and obsequiousness to the Herodian kings and the Romans. Even the once-pious Hasmoneans had eventually become infiltrated by the Sadducees. They'd fomented unnecessary wars, sown discord among the Jewish people (see Talmud, Yoma 9b), and had even killed a number of the leading Torah-sages (Talmud, Kidushin 66a). The Second Destruction, as painful and tragic as it was, at least accomplished the disappearance of the impious cliques (as alluded in the parable in the Talmud, Gittin 56b, in which the "barrel of honey" represents the Temple, and the "דרקון" hints to the Sadducees, among others). Once the Temple, and Jerusalem, were razed, the men of power melted away and the internal life of the Jewish communities returned to the aegis of the Torah-sages. Other non-traditional groups such as the Samaritans and the Essenes also now dissociated themselves from the Jewish people.
2) The political trigger to the Second Destruction was the anti-Roman revolt fomented by the Zealots, who acted against the advice of the Rabbis and left the Romans no choice but to crush the people (Talmud, Gittin 56).