It’s not because there are infinite decimals between every two decimal numbers. That applies to the rational numbers too, e.g. there are infinite rational numbers between 1/2 and 3/4. Rather, the real numbers are more dense in a way that makes them fundamentally larger than the integers / rational numbers. “Larger” means being able to pair up the two sets one by one so that each element of both sets is the member of a pair. No matter how you pair up the integers to the reals, you can prove that some real numbers will be unpaired.
You can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.
Oh right this is only true for irrational and transcendental numbers
Maybe this is misguided cheat, but couldn't you map any real number (between 0 and 1) to a natural number by mirroring the decimal digits across the decimal point. So 0.123 -> 321, but also sqrt(2)/2 -> ?601707 where ? is the rest of the decimal representation. This creates infinitely large numbers, but it's still a 1-to-1 mapping.
For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity.
In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something)
https://en.m.wikipedia.org/wiki/Aleph_number