Here I'm using the definition that an H-space is a topological space with a continuous map $ μ : X × X → X$ and a homotopy identity element $e$, i.e. the maps $μ(e, x)$ and $μ(x, e)$ are homotopic to the identity map.
I tried to show that the action of $\pi_1(X,x_0)$ is trivial on $\pi_n(X,x_0)$, i.e. if $[\gamma]\in \pi_1(X,x_0)$, then for every $[f]\in \pi_n(X,x_0)$, $\gamma_*([f])=[f]$.
How can I construct a homotopy between $f$ and $\gamma_*([f])$? I'm not seeing it. Even just pictures/drawings of the homotopy would help.
There is a more homotopical way of defining the action of the fundamental group of $X$ on $\pi_n(X)$. More generally for any CW complex $X$ with basepoint $x$ we can define an action of the fundamental group of $(Y,y)$ on $[X,Y]_*$ by saying a loop in $Y$ acts on a map $f$ by extending the loop into a homotopy of $F:X \times I \rightarrow Y$ from $f$ to $f'$ by the homotopy extension property and then defining the result to be $f'$. You can read more about this in section 4A of Hatcher.
This action coincides with the typical action when $X = S^n$.
Using this definition it is more or less straight forward. Suppose I have a loop $\gamma$ in my H-space $X$ and a map $\alpha :S^n \rightarrow X$. Then I can define a homotopy $\Gamma : S^n \times I \rightarrow X$ by $\Gamma(s,t)=\alpha(s)\gamma(t)$ this is a homotopy from $\alpha$ to itself where the basepoint follows the loop $\gamma$. Thus the action is trivial.