This is good, but I think it's cyclical (heh). We want to compare our signal with a "pure tone". What's a pure tone? A sine wave. Why is a sine wave a pure tone? Because when we compare it with a pure tone, it's identical.
LTI systems closely model many practical systems, including the tuning forks and flutes that give our intuition of what a "pure tone" means. I guess there's a level after that noting that conservation laws lead to LTI systems. I guess there's further levels too, but I'm not a physicist.
That eigenfunction property means we can describe the response of any LTI system to a sinusoid (again, complex exponential in general) at a given frequency by a single complex scalar, whose magnitude represents a gain and whose phase represents a phase shift. No other set of basis functions has this property, thus the special importance of a Fourier transform.
We could write a perfectly meaningful transform using any set of basis functions, not just sinusoids (and e.g. the graphics people often do, and call them wavelets). But if we place one of those non-sinusoidal basis functions at the input of an LTI system, then the output will in general be the sum of infinitely many different basis functions, not describable by any finite number of scalars. This makes those non-sinusoidal basis functions much less useful in modeling LTI systems.
Here, I just tried to define them “Pure tones are signals that have the unique property of putting all of the energy at exactly one frequency” and then use sine as an example.
Truly, complex rotations are a much better definition, but it’s somewhat reasonable to expect people to be familiar with sine waves.