When you learned that apples are composed of atoms, did you also conclude that apples are just an illusion, so there is no point in eating them?
I guess the answer here is that apples are still real on a macroscopic level, and everything you learned about them remains true. On a microscopic level, there are just configurations of particles. These particles, in some configurations, create what we perceive as an apple.
To understand the composition of the apple, you have to divide it into smaller parts. Those smaller parts are not apples.
Now apply the same reasoning to time.
For all the usual purposes, time is real. If you want to understand "how the time was built", you have to find the underlying mechanism, and whatever that mechanism might be, it is not time. (Because, generally, the explanation for X is not X. Otherwise we would call it circular logic.)
For example, we could model time (ignoring quantum physics) as a set of "possible moments", which are connected by arrows according to the laws of physics. In this model, the "flow of time" would kinda mean moving your finger along the arrows, starting from an arbitrary point, and watching how the situation evolves... except that from your perspective, nothing really evolves, all those possible moments are frozen, it's just the movement of your finger that points at different moments.
The obvious question in this model is: given that the laws of physics are reversible on small scale, why do these moments "contain information" about their past, but not about their future? We could just as easily move the finger in the opposite direction.
One problem with this model is that it ignores quantum physics. If you add it to the model, you have multiple arrows going from one moment into the future. Moving your finger along the arrows means you need to choose randomly (the Copenhagen interpretation) or you need many fingers (the Many Worlds interpretation). This seems to give some answer: the past is unambiguous, the future is not.
Only, this is still not a correct model of quantum physics. In quantum physics we don't really have one state evolving into a collection of states, but rather a collection of states (with a complex amplitude each) evolving into a different collection of states (with a complex amplitude each). So in the metaphor of the finger moving between the moments, each "moment" would actually refer to something quite complex. And it would get even more complicated if we tried to add relativity to this model (what is a "moment" if there is no absolute time frame?).
Also, this model still feels wrong to me. If we imagine the set of all "possible moments" (possible configurations of particles in the universe), and then select anthropically for those where humans exist... then our universe still seems much more regular than a random selection of these moments. (People like to talk about Boltzmann brains, but I haven't heard a good explanation why a Boltzmann brain remembering an orderly past should be more likely than a Boltzmann brain remembering a chaotic past; it should actually be the other way round, isn't it?) Which means I am still fundamentally wrong about something... not sure what exactly.
Anyway, my point is that instead of calling time an "illusion", it would be proper to talk about time being composed of something that is not-time. (I don't know what that something might be, and I have no idea whether anyone does, but the simple models I have heard about are all wrong.) The fact that time is composed of something doesn't make it less real, just like apple being composed of atoms isn't less of an apple.
The psychological arrow seems to be mostly about memory (we remember the past, not the future), which is about information (human memory is just another information storage medium), which I guess is related to quantum entanglement and possibly some other things (such as our universe containing lots of useful energy). But this is not specifically about humans; the iron also rusts only in one direction of time.