I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram
then it says the reader should supply the elementary geometry necessary to derive the homotopy
But I don't know how to do that. Please show me how to derive the above homotopy in detail so that I could try the identity and inverse properties by myself.
I am not in favour of restricting to paths as maps $[0,1] \to X$. Think of a path as a journey.
Instead you can take a path of length $r \geqslant 0$ to be a map $[0,r] \to X$. Not surprisingly, a path of length $r$ composed with a path of length $s$ gives a path of length $r+s$, and this composition is associative. You also have paths of length $0$, of course. Thus the paths on $X$ now form a category, say $P(X)$.
The question of the appropriate equivalence relation on $P(X)$ to give the fundamental group(oid) has to be discussed, and also reparametrisation to get a path of length $1$ from a path of length $r$. An account of this approach is in the book T&G= Topology and Groupoids. The classic book by Crowell and Fox on "Knot theory" also uses paths of length $r$, with a different but equivalent approach.
When I was writing the first edition of T&G in the 1960s I found several examples in the literature of complicated formulae with accompanying diagram, and I worked hard to get an account which avoided them as much as possible, and which I could understand. On one occasion this led to an entirely new result, a gluing theorem for homotopy equivalences. Groupoids came in when I tried to get a new proof that the fundamental group of the circle is the integers. See the discussion at https://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one.
Just to add a point: if one takes an aesthetic view of mathematics, and that one is looking for proofs which make things clear, then one look at the formula given for $H(t,s)$ should make one feel that there must be a better way.
In 1958 I overheard a comment of Raoul Bott on Alexander Grothendieck, part of which was that "Grothendieck was prepared to work very hard to make arguments almost tautological".