Computers are binary, yeah? So we have to represent fractional numbers with binary, too.
In decimal, numbers past the decimal point are 10-1, 10-2, ... etc. In binary, they're 2-1, 2-2, ....
2-1 is one half, so 0.1 in binary is 0.5 in decimal. 2-2 is one quarter. 0.11 in binary is 0.75 in decimal. And of course you've got 0.01 = 0.25
The problem comes when representing decimal numbers that don't have neat binary representations. For instance, 0.1 in decimal is actually a repeating binary number: 0.0001100110011...
Infinities do have different sizes, yes. But not on that scale. Both of these are countably infinite sets.
Think about this: there are infinitely many primes. Obviously, not every number is prime. But you can still map primes 1:1 with the natural numbers. They're both the same size of infinity.
This happens all the time. I feel like a big reason people don't like meetings is that they tend to involve a lot of bikeshedding.