So this is complicated. My immediate source for much of this is "The First Crusade" by Thomas Asbridge, but I've also done several courses on the crusades as part of my degree.

Short answers:

Normal people outside Jerusalem and its immediate surroundings didn't care very much and some came to like Christian rule. Islamic rulers did not raise big armies because they were more concerned about fighting each other and realised that trading with Europe could be very profitable.

Long answers:

I'll start with the question of Islamic rulers. In the 10th and 11th centuries the unity of the Caliphate had crumbled and rival Sultanates competed for power and religious legitimacy. Like Europe, which was split between two popes, the Middle East was split between two caliphs. The Sunnis, led by the Abbassid Caliphate, controlled Asia Minor and Syria and was based in Baghdad, whilst the Shiites, led by the Fatimid Caliphate, controlled Egypt and were based in Cairo. Added to this was the arrival of the Seljuk Turks. These horse riders swept into Anatolia from the Caucuses and battered the Sunni rulers in northern Syria and Anatolia. This meant that there were three major groups, all hostile to each other, engaged in constant military operations against each other.

To make matters more complicated for Muslims in the region, there was a string of deaths among the leadership of almost every major Muslim territory, as well as both caliphs. This threw the entire region into a mess of sultanates and city-states such as the Sultanate of Rum - led by Kiilij Arslan - Aleppo, led by a man named Ridwan, and Damascus, which was led by Duqaq. They were so preoccupied with each other that even if they knew of the impending military expedition from the west, they had bigger concerns or were simply too small to mount a defence.

When the crusaders reached Jerusalem it was decently defended by the garrison, but there was no Fatamid army nearby and they were caught off guard. The Fatamids did send an army when they heard about the crusaders reaching Jerusalem, but it was defeated and relations were normalised soon after. Surrounding states were generally ok with the presence of Christian settlers, at least initially, and they opened up trade relations that proved to be very profitable. We are fortunate enough to have the autobiography of Usamah Ibn Munqidh, who negotiated trade and alliances on behalf of Damascus for a time, as a source on this.

Now onto the people. This varied town by town, but the Gesta Francorum gives an account of the march from Antioch to Jerusalem. It noted that the Sunnis of the Holy Land (who were the overwhelming majority of Muslims there) were generally friendly to the crusaders because the region was ruled by the Fatamids, and it was hoped that the crusaders would be nicer to them than the Fatamids, who levied very heavy taxes. The crusading army was able to march from Antioch to Jerusalem in a month because towns opened their gates and set up markets. An expcetion was the town of Maraat, which was not cooperative. The crusaders massacred the entire town as a response. Raymond of St. Giles also attacked the walled town of Arqa - which wasn't even on the route - for no stated reason (though by the fact that he later returned to the region to establish his own kingdom, he was probably after territory for himself) and failed spectacularly. These two episodes soured relations, but not severely enough for people to flee en masse since the crusaders could still find markets.

Many sources - Ibn Jubayr (an Iberian pilgrim), Usamah Ibn Munqidh (a diplomat) amd William of Tyre (bishop and settler) - all noted with surprise that in the first few decades of the Crusader States, the local Muslims and the Frankish settlers were getting along well and integrating. The settlers learned Arabic and some adopted local fashion and intermarried. Among the reasons identified for this were the lower taxes of the Christian rulers, the abundance of trade, and a mutual interest in each other's culture.