Motivation for the one-point compactification

What we actually care about is relationships between topological spaces, and "$A$ is the one-point compactification of $B$" happens to be a particularly nice relationship about which it is possible to say a lot. For example, the sphere $S^n$ is the one-point compactification of $\mathbb{R}^n$, and this observation makes it possible to prove things about $S^n$ by passing to $\mathbb{R}^n$ or vice versa.

Example. The sphere $S^n$ is simply connected. One way to prove this is to show that a path in $S^n$ can be deformed so that it misses one point (this is the hard step). From here, removing the missed point gives a path in $\mathbb{R}^n$, which can be deformed into a constant path using linear functions.

In addition, a general principle in mathematics is that things which are unique are probably important. The one-point compactification is a construction of this type: it is the unique minimal compactification (of a locally compact Hausdorff space).