There are different ways to compare the "sizes" of infinite sets. So you could both be right in different contexts and for different sets. But the one concept people mostly mean, when they say, that some infinities are larger than other, is one to one correspondence (also called "cardinality"):

If you have a set and you can describe how you would choose one element of a second set for each element of the first and end up with every element chosen, than that's called a one to one correspondence. In that case, people say the two sets have the same cardinality which is one way to define their size (and a very common and useful one).

For example there is a one to one correspondence between the integers and the even integers. The procedure of choosing is to just take the integers and multiple each of them by two. You always get an even number and take that one to correspond to the integer you started with. So these two sets have the same cardinality and in that sense, the same size.

There is even a procedures that proofs, that the set of the rational numbers has the same cardinality as the natural numbers.

But Cantor proved, that there can never be such a procedure, that establishes a one to one correspondence between the natural numbers and the reals. So it's in that sense, that people say the reals form the larger set.