Different factions responded very differently. Some were very concerned, others had bigger problems to deal with, a few were (initially) pretty happy about it.

The first Muslim ruler the crusaders faced was Kilij Arslan, a Turkish sultan who ruled most of Anatolia. He was at war in eastern Anatolia at the time, and initially dismissed the crusaders as a threat. Before the proper crusade showed up, a highly disorganised force known as the 'People's Crusade' had attempted to enter Anatolia and been crushed by one of Arslan's generals. The People's Crusade was more of a rabble than a military force. It had no agreed upon commanders and leadership shifted as different individuals gained and lost the confidence of the crowd. There were ethnic tensions within it, with French and German crusaders attempting to outdo one another, and it had no plan. They were defeated by a small force gathered from local garrisons. So when the First Crusade proper showed up, he didn't do anything. He thought, based on this prior experience, that they'd be defeated by the garrison of the first town they came across. This abruptly changed when the First Crusade attempted to conquer Nicea and won. He turned his army around and headed west to meet this force in person, which he did at the Battle of Dorylaeum on 1 July 1097. The Turks fought like hell (which the crusaders admired them for) but ultimately lost the battle. In military terms, the defeat probably wasn't that bad, but with so much on his plate Arslan made the decision to just leave the crusaders alone, preserve his forces, and live to fight another day.

After making their way across Anatolia the crusaders arrived at the city of Antioch by the eastern Mediterranean coast. The governor of Antioch, Yaghi-Siyan, was aware that the crusaders' path was taking them to his city, so he sent out letters to various local noblemen asking for help. By this point, every ruler in the Middle East must have found out, if they didn't already know from the whispers of traders or diplomatic communications with the Byzantine Empire or Kilij Arslan. However, the Seljuk Empire, which had conquered everything from Georgia in the north to Jersusalem in the south, was fragmenting. This resulted in regional wars and conflicts of interest between the various Muslim rulers of the region. Everyone in the Seljuk Empire was too busy with regional conflicts and jostling for power to form a unified response in time, and the Siege of Antioch dragged on for months before a relief army arrived. The first army that attempted to relieve Antioch retreated after running into a force of knights, as the commander was not invested in the outcome and left once the excuse was there. A second, much larger relief force was assembled but arrived too late to prevent the city's fall and, when it did fight the crusaders, it suffered the same command issues as this was an even larger coalition of leaders. The second time was even worse, as the leader of the army (Kerboga, Atabeg of Mosul) was actively undermined by his new 'allies', who plotted to retreat early and leave Kerboga's forces alone on the battlefield. Particularly noteworthy was Duqaq, leader of Damascus, who was the first to flee and who capitalised on the defeat of the Muslim army by taking control of Homs as the other leaders were licking their wounds. The failure to relieve Antioch tells us that, while some were concerned about this western army, others were indifferent or saw opportunity in the chaos it could create.

So the leaders of the Seljuk Empire were too divided to form a good response. Another major Muslim faction, the Fatimid Caliphate, was actually quite happy about the arrival of the crusaders. The Fatimids and Seljuks had been at war for years and were stuck in an awkward stalemate. To the Fatimids, the crusaders were the key to breaking that stalemate. While the crusaders were at Antioch, Fatimid envoys arrived to propose an alliance, initially believing them to be mercenaries for the Byzantine Empire. Their offer was actually pretty sweet: the crusaders could take anything they wanted from the Turks and keep it, and the Fatimids would coordinate attacks into Seljuk lands. Their price was Jerusalem. The Fatimids had taken advantage of the Siege of Antioch to march north and claim the holy city, and they expected to keep it. The crusaders, who had marched thousands of miles, lost thousands of people, and endured unimaginable hardship all to reach and conquer the city, refused. Obviously, the Fatimids then knew the crusaders' intentions and prepared to defend Jerusalem, though their reinforcing army arrived too late to protect the city and was defeated at the Battle of Ascalon.

So the Muslim rulers of the Middle East had advanced warning that an army of Europeans was coming their way, and many knew its purpose of reaching Jerusalem, but they formed no unified response because... why would they? The elite of the Seljuk Empire were not popular, and the enemy of an enemy could be a friend. Many Muslim rulers saw the First Crusade not as a threat, but as an opportunity, so all attempts to create a unified response were undermined by those who thought they could profit from a new faction in the region, or could not overcome their personal rivalries to actually work together.

Adding to what u/J-Force has already said, there are very few, if any, contemporary sources about the first crusade from the Muslim side. It seems that most of them didn't realise the true scope or the intention of the Christian host and instead thought of them as mercenaries employed by the byzantines.

Yaghi Siyan did appeal to the emirs of Aleppo and Damascus (both of whom approached at separate time due to their rivalries and were beaten back) and to the Caliph in Baghdad, who dispatched Kerbogha. Afterwards, when the crusaders advanced to Jersualem, the fatimids were definitely aware of it and prepared for it by poisoning local wells and expelling the Christian population from the city. Al-Afdal, the vizier of the Shia Caliphate was also preparing a relief force, which arrived a little late but was defeated at the battle of Ascalon.

After the crusade a muslim jurist from Damascus, Ali ibn Tahir al-Sulami, seems to have correctly identified that the Franks were waging a holy war targeting Jerusalem but his calls for action against them weren't heeded. (Source: Asbridge, The Crusades)

Regarding your follow-up question regarding the intentions of the first crusaders, many of them also harboured personal ambitions. Bohemond for example refused to follow the crusade to Jerusalem and held onto Antioch as his prize.

Edit: Regarding some books for Islamic perspectives I would suggest Crusades through the Arab Eyes and Thomas Asbridge- The Crusades. Asbridge shifts to muslim and Christian perspectives with almost each chapter and is unbiased. He also mentions multiple contemporary islamic sources.