Mostly it was for practical reasons - why destroy perfectly useful buildings? The crusaders could reuse them for something else, rather than destroy them and have to spend time and money building something new. Also, either out of ignorance or by intentionally creating a sort of “pious fiction”, they associated the Temple Mount with ancient Biblical sites, rather than Muslim constructions.

Just to briefly recap, the two main buildings on the Temple Mount are the Dome of the Rock and al-Aqsa Mosque. They're located on the same site as the Temple of Solomon (destroyed by the Babylonians in 587 BC) and the Second Temple (destroyed by the Romans in 70). The Romans may have built a temple to Jupiter there, but after the empire was Christianized, the main religious centre of the city moved to the Church of the Holy Sepulchre, and the Temple of Jupiter was removed (if it existed at all). When the Muslims captured the city in 638, the Temple Mount was uninhabited and filled with rubble. Around 692 the Dome of the Rock was completed. It’s built around a rock, the Foundation Stone, that is supposed to be various things - the place where the world was created, where Abraham was going to sacrifice Isaac, also the Holy of Holies where the Ark of the Covenant was kept.

The al-Aqsa Mosque was also built around the same time. Muslims consider it to be the site that Muhammad visited on his “night journey” to the “farthest mosque” (“aqsa” = farthest).

The crusaders believed (sincerely or not) that the Dome of the Rock and al-Aqsa were two different ancient temples. The Dome of the Rock = the “Temple of the Lord” (Templum Domini) and al-Aqsa = the “Temple of Solomon” (Templum Salomonis), even though it was well known from Biblical history that, although there were two temples, there weren’t two separate ones that existed at the same time, and in any case both had been destroyed. Some crusaders apparently did believe they were the Biblical temples, but most of them understood what they really were. For example, William of Tyre, the court historian of the Kingdom of Jerusalem in the 12th century, explained that both were built by the caliph Umar. Apparently he was even able to read the Arabic inscriptions in the Dome of the Rock (or someone translated them for him, maybe).

So, why did they really keep them intact?

“The Franks would not at that time have had the financial resources to replace the architecturally remarkable structures on the Temple Mount with worthy Christian buildings. They may also have been aware that the destruction of two of the most important holy buildings in the Muslim world might bring about a consolidation of the Muslim forces against the Christians, the lack of which had allowed them to take Jerusalem.” (Boas, pg. 90)

Instead, al-Aqsa was used for various purposes - at first it was the royal palace, but it became rather dilapidated under King Baldwin I, who removed and sold the lead from the roof. In 1119 it was handed over to the Knights Templar, who used it as their headquarters. In fact that is where their name comes from - they were the knights of the Temple of Solomon. The Templars built a few other small chapels there (my favourite is the Chapel of the Cradle, where Christ’s baby cradle was supposed to have been located), and they used the ancient underground vaults as stables. They also built a new wall to mark out their territory on the southern edge of the Temple Mount.

The Dome of the Rock was converted into a church, and the crusaders changed the structure a little bit - they covered the Foundation Stone with marble and an iron fence to prevent pilgrims from chipping away at it for souvenirs. They added an altar and painted some frescoes on the walls. The Dome had been gilded (as it is again today), but the crusaders removed the gold and replaced it with lead, so it wouldn’t outshine their main church (the Holy Sepulchre). But there was still some gold there because a golden cross was added to the top. Elsewhere on the Temple Mount, an entire monastery was built for the Augustinian monks who settled in Jerusalem. They also built houses and gardens there.

Pretty much all of this stuff was removed or destroyed by Saladin when he recaptured Jerusalem in 1187. He turned

“mosques that had been converted by the Franks into churches back into mosques, by removing the church furnishings and erasing the structural changes made to these buildings, and by converting other structures built by the Franks into mosques and madrasas. He tore down the gilded cross from the Dome of the Rock and dismantled many of the Christian structures on the Temple Mount, including the monastery of the Augustinian canons which was located to the north of the Templum Domini (Dome of the Rock).” (Boas, pg. 16-17)

This probably isn’t that unusual for the medieval period - some other examples are the mosque of Cordoba, which was turned into a cathedral, and Hagia Sophia, which was turned into a mosque. Even the Parthenon in Athens was turned into a cathedral, and then into a mosque under the Ottomans. Sometimes buildings were destroyed; aside from Saladin’s destruction of the Augustinian monastery, the Ottomans also knocked down some of the other churches in Constantinople. But usually no one wanted to destroy a perfectly good building that could be used for new purposes.

Source:

Adrian J. Boas, Jerusalem in the Time of the Crusades: Society, Landscape and Art in the Holy City under Frankish Rule (Routledge, 2001)

edited to add John Giebfried, "The Crusader rebranding of Jerusalem’s Temple Mount," in Comitatus: A Journal of Medieval and Renaissance Studies 44 (2013).