How to show the fundamental group of torus is abelian in a homotopic way?

Draw a picture.

If you know how to visualize the universal cover of $S^1 \times S^1$, then it will become very clear how to visualize the homotopy. The universal cover is $\mathbb{R}\times \mathbb{R}$. Each lift of the path $a$ is a horizontal path from $(m,n)$ to $(m+1,n)$ for some integers $m,n$. Similarly each lift of the path $b$ is a vertical path from $(m,n)$ to $(m,n+1)$ for some integers $m,n$.

So, each lift of $aba$ is a "horizontal-vertical-horizontal" path, and each lift of $a^2 b$ is a "horizontal horizontal vertical" path. If you start both of these lifts at the integer lattice point $(0,0)$ then they will both end at the same point $(2,1)$.

Now you have two paths in $\mathbb{R}\times\mathbb{R}$ with the same initial endpoint and the same terminal endpoint, and it should be pretty clear how to construct a homotopy between them in $\mathbb{R}\times\mathbb{R}$. In fact, you ought to be able to use the intuition gained from the picture to write down an actual formula for the homotopy.

And once you have done that, just project back down to the torus using the universal covering map $\mathbb{R}\times\mathbb{R} \to S^1 \times S^1$.