I don't have a background in functional analysis so I find this hard to understand. How could it possible that each "dyadic" dilation of a function f(x) (where the dilation is f(2^i (x))) forms a space of its own?
That is like saying, sin(ax) and sin(x) belongs to different spaces, but for me, they belong to the same space because they are form a family of functions . How should I conceptualize the relationship between a function and its space?
Can someone motivate this concept through linear algebra?
No contradiction. It is correct to say that they belong to different spaces, and thay they belong to the same space. There are enough linear spaces around to make all such statements true.
Here is an example from linear algebra:
Question
Do the vectors $(1,0,0)$ and $(0,1,0)$ belong to the same space or to different spaces?
Answer
Both are true. They belong to the same space $\mathbb R^3$. They also belong to different spaces $U = \{(x,0,0): x\in\mathbb R\}$ and $V = \{(0,y,0): y\in\mathbb R\}$.
Similarly in the quoted subspace. There is a big space $L^2$ which contains all of the functions under consideration. It contains subspaces $V_j$, $j\in \mathbb Z$. These spaces have the peculiar relation
$$f\in V_0 \iff f(2^j\cdot) \in V^j \tag1$$
Formula (1) is not a statement about some special function $f$; it is a description of the relation between $V_0$ and $V_j$.
Unfortunately, the multiresolution structure is inherently infinite-dimensional; I cannot demonstrate it on a finite-dimensional example.