This section on Homotopy Equivalence is from Wikipedia.
I'd like to know how the definition of X being homotopy equivalent to Y implies the idea that X and Y can be transformed from one to the other.
Take for example the Mobius strip and the untwisted strip. Following the definition, my understanding is that the two spaces are both homotopy equivalent to a circle, hence homotopy equivalent to each other. Indeed I can find maps between those spaces that satisfy the conditions in the definition. So if I follow the definition, it's all good.
Yet from there I have trouble seeing the visual transformation between the two spaces. Say we embed the Mobius strip in 3-dimensional Euclidean space. To go from the Mobius strip to the untwisted strip, for example, I imagine the Mobius strip first having to shrink into a circle and then expand to the untwisted strip. It's the "expanding" step that I don't get, since in this step a point of the circle must be mapped to multiple points of the untwisted strip. This doesn't sound like a proper function to me?
No, you don't transform one space to another this way. That is just the "intuitive picture" to guide you. What you have is a homotopy that takes the original starting position $x$ and time $t$, not a function from $H(x,t)$ to $H(x,t+\delta t)$, and it isn't interpolating from $X$ to $Y$ but $X$ to $X$ (for $gf\sim 1_X$) and similarly $Y$ to $Y$.