I think you should not try to tackle understanding "infinity" head on. It's literally too much for human mind. So I'll try to give an "intuitively" understandable answer, thus avoiding "infinity" from now on:
All the numbers in [0, 2] can easily fit into [0, 1]! We can play a game to show this:
You provide a number from [0, 2] and I'll return a number from [0, 1]. For any different numbers, I'll lose, if I fail to provide different numbers. I can only win, if I convince you that I won't lose.
You may start with 2.
I'll go with 1.
square root of 2! (square root of 2) divided by 2.
Zero, got ya. Nope, zero, too.
Any number. Half that number.
See, what I'm getting at? There are as many numbers in [0, 1] as there are in [0, 2], because I can define a method to get one for every one you provide. I'll assume this is convincing enough and I won the game.
So to measure something that is too huge to measure, you just compare it, piece by piece, to something about the same size. Well, actually you don't compare, but tell how you would compare. If you can find a way to compare, piece by piece, it's not bigger.
The problem is to find a way to compare piece by piece. The only help I can give you in this, is the advice to avoid confusing yourself by the-bad-word. Instead ask yourself: "Where does the huge number of pieces come from?"
[0, 2] looks twice as big as [0, 1]. But it's not, if you half it!
[0, x] looks x times as big as [0, 1]. So as long as you can divide by x, everything is fine.
We all know that dividing by zero isn't allowed. So we found something that's "smaller": [0, 0] has only one element, so it's clearly smaller.
I disallowed myself from using the-bad-word. So I need to put it like this: If x keeps growing. What will happen to say 1/x?
At some point
1 divided by ever-growing-x will be indistinguishable close to 0. And if you can't distinguish between 0 and
1/ever-growing-x the piece to piece comparison fails. This makes [0, ever-growing-x] "bigger" than [0, 1] or [0, not-growing-x].
The next step would be to have a look at y-growing-way-faster-than-x. But I guess that'd overdo things.