A "quick" "summary" by Feser is:
1. The Principle of sufficient reason (PSR) holds that there is an explanation for the existence of anything that does exist and for its having the attributes it has.
2. If PSR were not true, then things and events without evident explanation or intelligibility would be extremely common.
3. But this is the opposite of what common sense and science alike find to be the case.
4. If PSR were not true, then we would be unable to trust our own cognitive faculties.
5. But in fact we are able trust those faculties.
6. Furthermore, there is no principled way to deny the truth of PSR while generally accepting that there are genuine explanations in science and philosophy.
7. But there are many genuine explanations to be found in science and philosophy.
8. So, PSR is true.
9. The explanation of the existence of anything is to be found either in some other thing which causes it, in which case it is contingent, or in its own nature, in which case it is necessary; PSR rules out any purported third alternative on which a thing's existence is explained by nothing.
10. There are contingent things.
11. Even if the existence of an individual contingent thing could be explained by reference to some previously existing contingent thing, which in turn could be explained by a previous member, and so on to infinity, that the infinite series as a whole exists at all would remain to be explained.
12. To explain the series by reference to some further contingent cause outside the series, and then explain this cause in terms of some yet further contingent thing, an so on to infinity, would merely yield another series whose existence would reman to be explained; and to posit yet another contingent thing outside this second series would merely generate the same problem yet again.
13. So, no contingent thing or series of contingent things can explain why there are any contingent things at all.
14. But that there are any contingent things at all must have som explanation, given PSR; and the only remanning explanation is in terms of a necessary being as cause.
15. Furthermore, that an individual contingent thing persists in existence at any moment requires an explanation; and since it is contingent, that explanation must lie in some simultaneous cause distinct from it.
16. If this cause is itself contingent, then even if it has yet another contingent thing as its own simultaneous cause, and that cause yet another contingent thing as its simultaneous cause, and so on to infinity, then once again we have an infinite series of contingent things the existence of which has yet to be explained.
17. So, no contingent thing or series of contingent things can explain why any particular contingent thing persists in existence at any moment; and the only remanning explanation is in terms of a necessary being as its simultaneous cause.
18. So, there must be at least one necessary being, to explain why any contingent things exist at all and how any particular contingent thing persists in existence at any moment.
19. A necessary being would have to be purely actual, absolutely simple or noncomposite, and something which just is subsistent existence itself.
20. But there can in principle be only one thing which is purely actual, absolutely simple or noncomposite, and something which is subsistent existence itself.
21. So, there is only necessary being.
22. So, it is this same on necessary being which is the explanation of why any contingent things exists at all and which is the cause of every particular contingent thing's existing at any moment.
23. So, this necessary being is the cause of everything other than itself.
24. Something which is purely actual, absolutely simple or noncomposite, and something which just is subsistent existence itself must also be immutable, eternal, immaterial, incorporeal, perfect, omnipotent, fully good, intelligent, and omniscient.
25. So, there is a necessary being which is one, purely actual, absolutely simple, subsistent existence itself, cause of everything other than itself, immutable, eternal, immaterial, incorporeal, perfect, omnipotent, fully good, intelligent, and omniscient.
26. But for there to be such a thing is for God to exist.
27. So, God exists.
For the details see the twenty pages of chapter five ("The Rationalist Proof") in his book Five Proofs of the Existence of God:
* https://www.goodreads.com/book/show/35592365-five-proofs-of-...
Also covered are: Aristotle, Neo-Platonic (Plotinus), Augustinian, and Thomistic.
(The above is the summary which I've transcribed. I think I've typed it out correctly, but there could be some typos.)