I think finding a one to one relation between elements of one set to elements of another set is a convincing way to know if they have the same amount of elements without counting them. Like if you have two heaps of rocks, you could make two lines of rocks where rocks are side by side and if both lines end together, both sets have the same amount of rocks.

Doing the same with infinite sets is useful because we can't count the elements, but might be able to find a function that associates each element of one set to one and only one element of the other. (Regarding the rocks example, every possible way to put them side by side is such a function.)

For example, there are as many positive integers as positive and negative integers because f(x) = (x % 2) * (-x / 2 - 1 / 2) + (1 - x % 2) * (x / 2) from 0 to ∞ gives 0, -1, 1, -2, 2, -3, 3, ...

Now regarding uncountable infinite sets like the real numbers, someone else in the comment showed that ]0, 1] has the same amount of elements as [1, ∞ [ using f(x) = 1 / x. And regarding what we're actually looking for, let's say [0, 1] has the same amount of elements as ℝ, I didn't really pay attention but this seems convincing.