The problem statement is: Suppose $X$ is a topological space with base point $x$. Let $\gamma_0:I\to X$ be the constant map $\gamma_0(s)=x, \forall s\in I$. Suppose $\gamma:I\to X$ is a continuous map with $\gamma(0)=\gamma(1)=x$. Define $\tilde{\gamma}:I\to X$ by $\tilde{\gamma(s)}:=\gamma(1-s)$.
Now I am trying to find a explicit based homotopy $\{f_t: I\to X\}_{t\in I}$ between $\tilde{\gamma}*\gamma$ and $\gamma_0$. I would like some help for getting started, a picture will be extremely helpful because I can't see what is going on.
As you ask for a picture, here is one taken from Topology and Groupoids.
Note that this account deals with paths as maps $a:[0,r] \to X$ where $r \geqslant 0$, which is called a "path of length $r$". Thus if a path $b$ in $X$ is of length $s \geqslant 0$ and their composite $a+b$ is defined then that is of length $r+s$. This makes the exposition easier; addition of such paths yields a category, in particular associativity holds. Also this does not restrict to loops at a base point.