I am almost finishing The Grounding for the Metaphysics of morals and I believe I have a pretty good grasp of what deontology deems moral and what it deems immoral, however, I don't understand why we should follow it,is it because of a divine command? Or something related to reason?
To my mind it is like Crowley's reason you should study 'magic' (by which he means, basically, interpersonal psychological manipulation). You will do it anyway, and you might as well do it efficiently.
You will expect other people not to require you to act in ways that do not generalize to others. And so will they. And you will have some kind of war about it. Why not cut out the middle-man and circumvent the war?
Kant sees this as a direct consequence of reason. His categories and other axiomatic contents lead him to the deduction that every intelligence will act, perfectly or imperfectly, on this duty to allow every intelligence the respect it is due, and will work to make others do so as well. Beyond that, he deduces that all other duties are epiphenomena of this one central duty. Doing what you will eventually do imperfectly anyway, then, as perfectly as is reasonable now, just makes sense.
But you can also get there by direct self-observation. Every child has an impulse toward disliking those who are unfair, whether they focus on unfairness to themselves or third parties. Even psychopaths with no sense of conscience are bothered by unfairness toward themselves, and by inconsistency. And few of us ever lose that.
The ability to live morally according to the categorical imperative is -- in Kant's view -- what defines us as humans. Humans are not solely amenable to the laws of physics. In contrast to (other) animals, they are legislators in their own case -- not with respect to their physical movements, of course, but their willing.
As laws are general and as there are many legislators -- as many as there are humans -- the law each of us puts his/her willing under must not interfere with anybody else' law. Hence the categorical imperative. So, according to Kant, living in line with the categorical imperative is fully living up to the standards or capacities of a human being. This leaves us with the choice of living a human life or the life of an (other) animal.
Man who decides to live as animal, however, is bound to be unhappy as happiness requires living to one's full capacities ("talents" as Aristotle called these). Thus, if our goal is happiness it is even irrational to be immoral.
All in all -- one has the choice but being immoral comes at the price of the inner conflict of "being human but living the life of an animal" alias "being irrational (if happiness is a goal)", according to Kant.
I don't see why we should follow it, is it because of divine command?
Well, if the Greek myth of Prometheus is correct; and our rationality, and our consciousness of our rationality is a spark from the divine fire, then possibly it is a 'divine command'.
or is it because of our reason?
I don't take it - and I might be very wrong here - that Kant is saying that one specifically in ones daily acts and actions must be Kantian; I take it he's providing an axiomatic, and therefore rational basis to moral behaviour and thinking; it's in the structure of rationality - so it's followed, in a sense, whether we note it or don't.
After all, though Newtons First Law is universally true; in our everyday experience the exceptions to this rule are always the case.
The shortest answer I can think of is a short text from the Conclusion of the Critique of Practical Reason:
Two things overwhelm the mind with constantly new and increasing admiration and awe the more frequently and intently they are reflected upon: the starry heavens above me and the moral law within me.
Though heavily discussed until today, it is quite secure to state that admiration (sometimes respect, like in GMM) is the reason why we act morally, though not our motivation: this feeling is described as being evoked by moral acting, not the other way round.
And Kant does think it as an anthropologic fact that we do want to act morally, see for example GMM, Ak. 454, where he states that even the most evil person, as long as she is used to think rationally, will want to be a better person.