A Look at St. Thomas Aquinas' Third Proof for the Existence of God - Summa Theologiae I, QII, AIII, I Answer That
Essence and Existence
In order to understand the strength of Aquinas' logic, it is necessary to first understand a few terms that he uses. These terms are not easy and are certainly not exhausted by the simple definitions that I am providing. Often times people fail to understand Aquinas' arguments because they do not spend the time plumbing the depths of meaning behind the terms that he uses. Nevertheless, we are going to give a first attempt here.
Essence Alone - Let's think of three categories of beings in a little thought experiment. First, let's think of realities where we can conceive of a type of thing, as "essence," even though that thing doesn't or hasn't ever existed. For example, we can think of a unicorn. A unicorn is like a horse, but has one long horn, and usually has magical powers. That is the essence of unicorn-ness, but regardless a unicorn will never exist. We can think of its essence, but its essence doesn't include real existence at all.
Essence AND Existence - Second, let's think of realities where we can think of a thing that can potentially really exist. A dog, for example. Dogs have four legs, a snout, a certain K9 disposition, etc. But differing from a unicorn, dogs really exist. Now, we can also think of dogs as not existing. There's no reason why dogs must exist. They could all go extinct if we are not careful. So this second category are essences that can have existence, but don't have to exist. Aquinas calls them, "potential beings."
Essence EQUALS Existence - A third category is necessary then. In this category we think of beings that by their definition must exist. They cannot, not exist. We can think here of the law of conservation of matter and energy, as an example. If something cannot not exist, then Aquinas calls it a "necessary being" by virtue of its essence and definition. He does make a distinction, though, between necessary beings through themselves and those who depend on others. We will come back to this below.
Now that we have some background on the terminology, we can move to the argument itself. It begins with an observation about the world around us that we can see. It then moves to a principle of how reality works, which presents a problem to be solved. Finally, Aquinas presents the only logical solution to that problem and a conclusion.
1) Observation - We see that around us that things can either be or not be. There are many examples of beings that are purely essence or only possible beings.
2) Principle - Possible beings, by definition, were at some point not in existence. If we go back far enough, then a possible being will not be in existence at some point.
3) Problem - Now if only possible beings existed, then at a point in the past nothing would have existed. If there was nothing, though, previously, how did everything come from nothing? Rather, logically, there should still be nothing if that was the case. There is definitely something in existence, though, clearly as we observe all around us.
4) Solution - To solve this problem, we must posit some type of necessary being, a being that by definition exists of its own accord, and which thus can be the source of all other beings. Can this be the material universe, though? We have already said that matter cannot be destroyed and thus it is a necessary being. Here is where that distinction above comes into play. Aquinas points out that in the first two ways we demonstrated that the universe does not have its necessity through itself. The material universe is subject to change and causality, and therefore the problem still remains ... How can we have a universe if everything in it does not exist by definition of its own essence, but can be thought not to exist?
Here we come to the final piece of the logical puzzle. There must be some necessary being which exists by its own essence, free from dependence on anything else. What type of essence would this have to be? This being would have to be a reality of pure perfection such that no perfection or existence could possibly be added. Only then would you have a being of pure necessity which could be responsible for the contingent existence of every other possible being.
This ultimate necessary reality of which all else is contingent upon, Aquinas says, is what we usually call by the name of God.
1 - Aquinas, Thomas. Summa Theologiae. Prima Pars, Q2, AIII, I Answer That. https://isidore.co/aquinas/english/summa/FP/FP002.html#FPQ2A3THEP1