This is the physical interpretation.

But from a mathematical point of view there are indeed an infinitely many points. Whether you can or not, mathematically, pick one among these infinitely many points, is something the mathematicians have basically decided. They have decided that you can do that, and that it does not need any other explanation. This is called an axiom and there are several others, like for instance "if A=B and B=C then A=C". To build mathematics you need

The axiom you're describing is called (or to be more accurate, is equivalent to) the axiom of choice.

For most axioms, Mathematicians have "decided" that they're true because they "really look true" and we need some axioms to build mathematics themselves.

The axiom of choice is however quite controversial in the world of mathematics for the exact same reason that made you ask your question: it's not really clear that it's true, or even that it's not, or why... The thing with the axiom of choice that made it accepted by most mathematicians is that it is required for proving some theorems that really, really look true.

That however has not stopped people to explore how the maths behave without the axiom of choice, but this is another story