Infinity doesn't make sense one bit. Add one to infinity and you get infinity. This breaks the mechanism of addition and if you include infinity as a number, math no longer makes sense because infinity leads directly to contradiction when used that way; no axioms hold up. However we can refer to infinity implicitly and this is what's done.
Say you have a pool of water with 1,000 gallons in it. Once every ten seconds, you dump exactly one gallon of water in, while your evil twin bails out exactly one gallon. Now over an infinite interval of time, does the total amount of water really change? No. So what we can do is look at clues as to what happens when things get near infinite, e.g. we know the volume of water doesn't change, and extrapolate.
Calculus does this very straightforwardly. If you want to know the area under a curve, you can't just measure it with like a ruler. If your need isn't exact, you can just estimate the area using straight lines and the area of each box added together gets close to the actual area underneath. As it turns out, if you generalize that process and reduce the size of each bit to infinity, you can get exactly the area underneath. We then learn a few tricks, so to say, to get the end result of this "integration" without having to actually sum up the area of boxes. This seemed very crazy when Newton and Leibniz did them: how the eff can infinitely small areas make a full area? Well, these "differentials", as they are called, proved to work in a huge amount of physics equations and helped tie together a disparate set of physics into nicely packed generalities. It really doesn't make sense intuitively, but the math works and is more well proved than the theory of evolution.
Then we get to where we need to describe numbers in a language that is very rigid. This is where set theory, and people like cantor come in. Set theory attempts to make sense of cantor's work. Cantor took the idea of infinity seriously, as wrote about it in length. His work is still preserved in set theory, though there's not much point in reading his work directly.
Why was cantor's work resisted? Well, the Greeks had a phobia for infinity a long time ago and Euclid's Elements was the primary source for teaching geometry for hundreds of years. The Greeks believed all numbers could be expressed as rational numbers, so pi was extremely irritating to them, as it's an irrational number that represents a comparison of radius to circumference. You also had Zeno's paradoxes which attempted to show infinite division was impossible, and I also remember there being a paradox about the edge of space being a contradiction, meaning they believed empty space to be infinite. All of this is in line with Euclidean geometry.
TL;DR Euclid introduced biases that simply held until calculus proved the use of infinitesimals and set theory was a necessary development in advanced mathematics