Infinite-dimensional vector spaces also show up in another context: functional analysis.

If you stretch your imagination a bit, then you can think of vectors as functions. A (real) n-dimensional vector is a list of numbers (v~1~, v~2~, ..., v~n~), which can be thought of as a function {1, 2, ..., n} → ℝ, where k ∊ {1, ..., n} gets sent to v~k~. So, an n-dimensional (real) vector space is a collection of functions {1, 2, ..., n} -> ℝ, where you can add two functions together and multiply functions by a real number.

Under this interpretation, the idea of "infinite-dimensional" vector spaces becomes much more reasonable (in my opinion anyway), since it's not too hard to imagine that there are situations where you want to look at functions with an infinite domain. For example, you can think of an infinite sequence of numbers as a function with infinite domain. (i.e., an infinite sequence (v~1~, v~2~, ...) is a function ℕ → ℝ, where k ∊ ℕ gets sent to v~k~.)

and this idea works for both "countable" and "uncountable" "vectors". i.e., you can use this framework to study a vector space where each "vector" is a function f: ℝ → ℝ. why would you want do this? because in this setting, integration and differentiation are linear maps. (e.g., if f, g: ℝ → ℝ are "vectors", then D(f + g) = Df + Dg, and ∫*(f+g) = ∫f + ∫g, where D denotes taking the derivative.)