I was wondering, is Islam the religion of reason? If Islam is the religion of reason, why would such intelligent people not believe in it? Is there a problem with Islam or with their intelligence?

Let's start answering with reason. Do we have a clear answer as to what exactly reason is? What is reason, is it the power of intellect or the capacity of mind? Let's elaborate a little on this. Is it grasping something immediately, or absorbing and retaining the information over time?

I am actually confused!

Well, the issue itself is complicated anyway. Reasoning is often equated with intelligence nowadays, and highly intelligent people are considered to be very logical. Intelligence, however, is defined as the power of a person to find solutions to problems in one's own field. Solving the problem is only possible by deep comprehension and retaining it permanently. Here the mind comes into play, it is where what we have learned is stored.

So does the mind do all this on its own? Doesn’t it get support from anywhere?

We can answer this by thinking about the development of a child. The child is a smart creature from the moment of birth as long as s/he doesn't have any mental illness or retardation. The first learning experience of the child is through eyes and ears. That is, children start learning by imitating the things they see and hear. The things that are said do not make sense to the child, and the child does not have the ability to reason. With the development of the mental ability over time, the child can remember the things that hurt them a lot. S/he may have no idea why that things hurt. This is because the power of comparison is not yet developed in the child. The children only reach the stage of using their minds with the development of the power of comparing and differentiating. Up until this time, a great deal of input accumulates in children's minds through what they see and hear, although this is through imitation. This is how the children make the comparison and differentiation. If it is insufficient, they tend to get information from those around them. Exactly at this point external sources come in.