The Temple Mount was probably a site of cultic sacrifice since pre-Israelite times, but the reason the site became so important in Judaism was because it was centralized as the only place where sacrifice could be offered. This centralized political and economic power in Jerusalem. The Temple was literally seen as God's "house" on Earth. Even now, at least in Orthodox belief,, Jewish sacrifices cannot be resumed until the Temple is rebuilt.

Jerusalem is important to Christians for the obvious reason that Jesus was crucified there and (in Christian belief) was resurrected there. This was a direct result of Jesus having himself been a Jew and having been in Jerusalem for a Jewish festival and (ostensibly) for having caused some kind of disturbance at the Jewish Temple.

I know less about the Muslim associations other than that Muhammed is traditionally believed to have ascended to Heaven from the Temple Mount. Muslims identify their God as the same as the Jewish God, though (the God of Abraham, Moses and even Jesus) and the Temple Mount is also traditionally (in both Islam and Judaism, I think) the site where Abraham was going to sacrifice Isaac.

It all goes back to Jerusalem being, for centuries, the political, religious and economic center of Judaism and of Christianity and Islam both being offshoots of Judaism.