While learning about homotopy in my Algebraic Topology course I (as someone who is at least aware of higher category theory) noticed that it's possible to define a notion of "homotopy between homotopies":
Let $f,g:X\rightarrow Y$ be continuous maps between topological spaces and $H,K:f\simeq g$ two homotopies from $f$ to $g$ i.e. $H$ and $K$ are continuous maps $X\times I\rightarrow Y$ such that $H(x,0) = > K(x,0)= f(x)$ and $H(x,1) = K(x,1) = g(x)$ for all $x\in X$.
Let $\psi: X\times I\times I\rightarrow Y$ be a continuous map such that for all $x\in X$ and $t\in I$, $\psi(x,t,0) = H(x,t)$ and $\psi(x,t,1) = K(x,t)$. Then $\psi$ can be considered as a homotopy $H\simeq K$ between two homotopies.
Assuming that this is well-behaved with respect to compositions and "morphisms of lower degree" (e.g. interchange law of natural transformations etc), it makes sense to consider the $(\infty, 1)$-category $\text{Top}$ whose objects are topological spaces, 1-morphisms are continuous maps, 2-morphisms are homotopies between continuous maps, 3-morphisms are these homotopies between homotopies I have just defined, and so on...
I asked my lecturer whether adding in these extra "higher homotopies" is useful for doing topology and whether it gives any extra interesting information. He said it is and does, but explaining why is a bit complicated! So, can anyone attempt to explain why we ought to care about these things reasonably simply?
The $(\infty, 1)$-category to really care about is the $(\infty, 1)$-category of (weak) homotopy types, which can be obtained from what you wrote down by restricting attention to spaces with the homotopy type of CW complexes. This gets rid of a bunch of pathological objects like the homotopy type of the Cantor set.
This $(\infty, 1)$-category, which I'll call $\text{Space}$, has the same relationship to $(\infty, 1)$-category theory as $\text{Set}$ has to ordinary category theory: namely, in the same way that an ordinary (locally small) category is a category enriched over sets, an $(\infty, 1)$-category is a category enriched over weak homotopy types. Furthermore, in the same way that $\text{Set}$ is the free cocomplete category on a point, $\text{Space}$ is the free homotopy cocomplete $(\infty, 1)$-category on a point.
At a more basic level, a general reason to care about giving things higher categorical structures is to make sense of higher categorical universal properties (such as homotopy colimits, alluded to above).