I What is the definition and use of homotopy in curve mapping?

[Woolyabyss]

I'm having difficulty understanding the definition of homotopy.
In the notes attached I'm not sure what the mapping does and why the cartesian product is used. does t represent the parameter and s any curve between phi(1) and phi(2)?
Any help would be appreciated

 

[Orodruin]

A curve is a map from [0,1] to D, where the parameter in [0,1] can be called t. The homotopy on the other hand is a map from two intervals [0,1] to D, with the two parameters now being called t and s, such that for s=0, the remaining map from [0,1] is one of the curves and for s=1 it is the other. This is what is meant by continuously deforming one curve to the other, each point (fixed t) along the curve makes another continuous curve, parametrised by s, and for each fixed s you have some continuous curve.