I don’t have the time to find all the sources (but they are somewhere in my previous replies to other threads), but the key to understanding the need for the first cause is to understand Aquinas’s distinction between essential (Latin: per se) and accidental (per accidens) causes.
The difference is not easy to grasp at first, but it boils down to the degree of dependence that the effect has on the cause.
Here are a couple of examples.
Pope Julius II wanted to have the Sistine Chapel redecorated, so in order to do that he needed a painter. He wanted it to be done really well, so he needed, not just any painter, but a top-notch painter.
Having a top-notch painter is essential (per se) to having a well-decorated Sistine chapel. You will get no paintings, or a lesser quality painting job, without one.
It turns out that Julius contracted Michelangelo Buonarroti. To get a first-rate chapel, he could have contracted Raffaello Sanzio da Urbino (“Raphael”) instead (the one who painted the Stanze Vaticane). Hence having Michelangelo as the painter specifically is not essential to a first-rate Sistine Chapel—it is accidental (per accidens).
More a propos to our topic, let’s look at something we are all familiar with: human ancestries.
Let’s think about what are the things that sustain us in our existence right now in this very moment; and to make things easier, let’s limit ourselves to efficient causes: food, water, the heat of the sun, the earth’s gravitational field, our shelter which protects us from the elements, and so on. (With a bit more investigation, we discern that our soul sustains our bodies’ continued existence; and God maintains the soul in existence; and so forth.)
There is an efficient cause that is notably absent from this list: our parents (or, generally speaking, our ancestors). Why not? Because, at the present moment, we do not depend on them (in the order of efficient causes, that is) for our existence. They exerted that type of causality once (especially at the moment of our conception, but also, in a different way, when we were growing in our mothers’ wombs, and also when our parents took care of us when we were children); but they do so no longer. For instance, when our parents go to be with the Lord, we will not at that moment cease to exist.
What this example illustrates is that causes that are remote in time cannot be essential causes; they are merely accidental causes. The effects of such causes are no longer dependent on them for their very existence (which is the definition of an essential cause).
Now, back to the infinite regress: what is not allowed is for there to be an infinite regress of essential causes. On the other hand (according to Aquinas anyway—and I think I agree with him), there is nothing preventing there from being an infinitely long chain of accidental causes, spread out indefinitely in time.