This is not "infty" as per computer representations of numbers. This is "infinity" as in "how many counting numbers are there" and "how many reals are there".

We are not dealing with floats. Trivial, for any size being used, the number of floats that can be represented is finite.

So what this is saying is this:

Consider the collection of counting numbers: 1, 2, 3, ... and so on. Call that set N.

There is a set P which has the property that every time you try to match up items of N with items of P, there are items of P left over.

If you have some cups and saucers, and every time you laid them out you had some saucers left over, you'd say you had more saucers. Yes? Well every time we try to match up things from N with things from P we can things from P left over, so we should say we have more of them, right?

But there are infinitely many things in N, so P must be a bigger infinity.

One simply cannot "count" with "real" numbers. But "floats" is probably countable since it's represented with limited digits. However, considering the inherent error associated with it, I wouldn't say it's countable. Also, if you are trying to find an infinite large number, for any arbitrarily large real number, you can find a larger integer and vice versa. But the point here is that there are more real numbers than integers.