One reason that operads are used is in obstruction theory. Suppose we have a CW-complex $X$ with basepoint and multiplication $\mu:X \times X \to X$ which is associative and unital up to homotopy, and we want to know if $X$ is homotopy equivalent to a topological monoid $X'$ by a map that, up to homotopy, respects the multiplication. This is a question about homotopy theory, but constructing a strictly associative multiplication is not amenable to methods of homotopy theory.

This is a question about the associative operad, but we can replace it by a question about an equivalent operad such as the collection of Stasheff associahedra. This has a simple presentation, as an operad, in terms of generators and relations, and so it is easier to classify actions on an object. This provides a sequence of obstructions in $\pi_k Map(X^{k+3}, X)$ to finding on an object (and the choices are similarly parameterized).

Of course, as with many things in algebraic topology, this general method works far better to show something does not admit a multiplication, or when the obstructions occur in zero groups. However, it is difficult to attack such problems outside specific circumstances without using operads. (Perhaps someone else knows of methods that don't implicitly use operads; I don't.)

I'm not complete clear on what constitutes the difference between "where" and "why" in your statement, but I hope this qualifies.