Some parts of mathematics deal with infinite sequences, that is, actually infinite lists of numbers. It's usually assumed, and important for analysis, that these numbers are considered to be "all there" right from the beginning. You can do operations like: Compute the limit. Add up all of its elements. Determine whether two sequences are identical for all entries after the trillionth.
I think this is often part of the misunderstanding when you stumble into a post by someone who's confused about 0.999... = 1. People sometimes write things like: "0.999... only moves closer and closer to 1, it never reaches 1." I think that highlights a deeper point than people usually give these comments credit for. The thing is, 0.999... doesn't "move" anywhere, it's considered a completed object right from the beginning.
Anyway, the point is that Turing machines are not like this at all. They only look at a fixed-size part of the tape during each step, from this follows that they have only used a finite amount of tape at each point of their execution.
So for any given (halting) computation, you don't actually need an infinite tape, you just need "enough", without changing the result. This is important because it makes Turing machines a model for practical computers. For example, the device you're reading this on has gigabytes of tape, and that's big enough for many, many, many kinds of computation.