Look up Cantor's Diagonal Argument
Sets are considered "equal" (don't know the english word for "równoliczne")
Imagine both infinities are "equal".
That means you can match every, say, natural number to a number between 0 and 1. So we assume you can list every rational and irrational number:
We match 1 with 0,183640..., 2 with 0,1947311....
etc., so that each real number is assigned to a natural number. That means the sets have the same anmount of numbers.
But now we can construct a number like this:
We take number one and change the digit at its first decimal point. So if it was 0,18... we take 0,2
Then we take the second decimal point from the second number and alter it
So if it was 0,194...
We take our original 0,2 and make it 0,28 (or 0,21, whatever. Just not 0,29)
We repeat the process forever.
The number we just got differs from every number on the list by definition:
If a number is nth on the list, our number has its nth digit different. Therefore, we just showed that the number we constructed is not on the list. We will always be able to create one like this. So, there are more real numbers between 0 and 1 than there are in the set of natural numbers.