Many operations on the homotopy groups of a CW complex can be realized by a "geometric" (rather, topological) construction on the space itself. A popular example is the rationalization, studied in rational homotopy theory.
The general idea is to write down a CW model for your space, with explicit attaching maps, and doing the construction on spheres, then re-glue by the attaching maps.
An easy example to look at is the following construction on the 1-sphere: take the covering $z \mapsto z^n$ from the 1-sphere to itself. A path on the "lower" circle that generates the fundamental group $\pi_1(S^1,1)=\mathbb{Z}$ can be multiplied with $n$ (going around $n$ times) and then lifted to the covering space and there it corresponds to the generator. Attaching a 2-cell along this covering map kills this $n$-times going around map on the homotopy level (since it is liftable, so contractible in the newly attached 2-cell). The result on the fundamental group is that you get $\pi_1(S^1 \cup_{z^n} e^2) = \mathbb{Z}/n\mathbb{Z}$.
You can iterate this with a mapping telescope construction to get an inverse limit over these groups, thus the fundamental group becomes the profinite completion $\hat{\mathbb{Z}}$ of the integers.
Since you asked for how to visualize this, I tend to draw pictures on the blackboard and imagine the paths and the contractions quite visually.
To address the question whether this is useful or just good to know it exists: Sometimes it can be crucial to know how many cells in which dimensions you need to attach to change the homotopy groups, and knowledge about the attaching maps can be important too. For example, from all attaching maps you can compute CW (co)homology, so you can look at how the (co)homology changes if you, say localize at a prime. Maybe the Quillen + construction is a good example for this, since one really has to attach only a 2-cell and then a 3-cell, to get it right.