Why is the operation $*$ well-defined on homotopy classes, and not all continuous paths from $[0,1]$ to $X$ in general?
I suppose "well-defined" means that if $a=b$ and $c=d$, then $a*c=b*d$. I feel this is valid for individual paths between points too. Why homotopy classes? What about non-homotopic paths?
Thanks in advance!
The usual path concatenation is well defined on paths whose end points match up - that is, we can evaluate $f*g$ if and only if $f(1)=g(0)$.
As to why we want to consider path concatenation as an operation on homotopy classes of loops (paths whose both end points are equal to the base point of the space $X$) when defining the fundamental group, or on homotopy classes of all paths when defining the fundamental groupoid, this is so that we get the corresponding structure. For instance, any non-constant loop concatenated with its inverse loop (the loop where you go the other way round) is not equal to the constant loop at the base point, however it is homotopic to the constant loop at the base point, and so path concatenation on homotopy classes gives us a group structure (we also need to consider homotopy classes in order to get associativity and existence of identities).