I many cases, probably not very much at all! At least not directly. The medieval middle east didn't have anything directly comparable to the European inn, which European travelers are known to have complained about considerably. Instead, they had structures known by names such as Caravansarais, Funduqs or Khans. These were often very large, architecturally impressive enclosed spaces where caravans could stop stop and rest and take on water. Their construction was often sponsored by the state or by religious institutions and they were often spots were cargo was inspected and taxes levied. They also tended to include or be next to bazaars were goods could be bought and sold, and served as a loading and unloading point for this kind of commercial activity. It might seem natural for these places to provide food as well, and it seems likely that in some limited cases they may have provided a little, but in general travelers would expected to take on more travelling food at the markets, eat stuff they'd brought with them or in the case of urban caravanserais, buy from the abundant street food vendors that crowded the streets. Remember that most medieval urban spaces didn't see a lot of home cooking, street vendors and public cookhouses were an important component of how people in general ate in these areas.
As for what these street vendors would serve, that is an expansive question since the middle east includes a vast range of different cultures with variable culinary habits. I'll just focus on Egypt since I know more about that. It was probably mostly stew, with the occasional roasted or baked good. Stew was, at this time, frequently the most common way to eat food across much of Eurasia so this isn't surprising. If you had more money it would likely contain meat, if you had less it would be mostly grains, herbs and pulses such as chickpeas and fava beans. Fermented fruits and vegetables and stuff like olives might be served as a relish. Higher end preparations were often elaborately spiced, with much the same spices used in the middle east today, and savory elements were provides through ingredients such as preserved lemons or murri, which ambiguously refers either to fish sauce or a soy-sauce like condiment produced from rotted barley. Dishes were frequently sweetened (medieval Egyptians definitely liked their food sweet!) and often included sugar, honey or dried fruit for that purpose. Fruit was likely available on its own in most cases, sometimes dried, sometimes fresh, as were nuts. In many cases fruit and nuts were pounded into paste like preparations called harisas and dried, and these considered excellent foods for travelers. Beans might also be pounded into products similar to modern hummus, but usually more elaborately flavored. Bread would often be served as part of meals, but rice was also popular, especially as you headed East towards Persia.
Some sources:
Lindsay, J.E., Daily Life in the Medieval Islamic World.
Lewika, P.B. Restaurants, Inns and Taverns That Never Were: Some Reflections on Public Consumption in Medieval Cairo
Lewika, P.B. Food and Foodways of Medieval Cairenes: Aspects of Life in an Islamic Metropolis of the Eastern Mediterranean