What you don't get? That's the way they behave. It is not different from any other chemical reaction. Why does the AgCl precipitate from the solution? Why does chlorine oxidize sodium? Why does carbon burn in oxygen? We have thermodynamical explanations, but they don't change the fact that it happens because it happens.
Then, we happened to call substances that donate protons acids. We could as well call them guripsestrudomes.
Sorry, but you don't make the question better defined by repeating it. As I already wrote - forces behind the acids donating protons are exactly the same ones that are behind any other chemical reaction. For some reasons products are more stable, or get separated (so that backward reaction is unlikely) and so on. Acid dissociation is in no way different.
Yes, I thought yesterday of answering in the same way - they are obliged to donate protons by definition. At least for about the last 80 years. Anything that doesn't donate a proton is not called an acid.
I held back because that would not have been very helpful, but from the chemist's point of view you did need, as Borek says, to ask a more meaningful question. Maybe better would have been: why is anything an acid? Which is again as Borek says a part of the question why does any chemical reaction happen?
You are asking us to write chapters of your textbook.
But roughly some bonds - the -O-H bond looms large, are weak enough that they will permit significant dissociation into -O- or rather, it is better to think of it as, transfer of a proton to water
X-O-H + H2O → X-O- + H3O+
Whereas there is no significant such dissociation of a C-H bond except for very special unusual cases.
(The removal of the proton has an energetic cost, but when it happens it is also because in compensation the charged species created attract and are attracted by water molecules that being dipolar, the oxygen atom having a partial negative charge and the H atoms a partial positive one) orient themselves around the created charges. This factor is not sufficiently emphasized in explanations, nor it it easy to calculate quantitatively.)
What you need to have an eye for, especially since such things as dissociation constants are very difficult to predict quantitatively, is the tendencies, how the acidity of molecules varies systematically according to: position of the relevant atoms in the periodic table, across and up and down, and the oxidation state of atoms. The explanation given to the main tendencies are succinctly summarised here.
In organic acids there are also, perhaps clearer, systematic tendencies to learn about and there are clear quantitative relations between acidity as tendency to donate a proton and other manifestations of electron density in other chemical reactions (read up about Bronsted and Hammett relations).
You will find these things quite a lot emphasized in chemistry teaching and examinations so it would do you some good to follow up your curiosity by reading up these things and having them present continuously as you go throughout things like chemistry of the elements, organic acids etc. Do not be surprised if not everything rationalises completely, I have hinted why.