I see that /u/metaphor has given you some formal references.

I'd like to chime in with a more intuition-based explanation of what transfer functions are, from my recollections of college control theory classes in both electrical signals and a more general "systems engineering" application:

Basically, the transfer function is a different perspective on modelling/representing a system's output as a function of its input. Classically, when modelling and/or reasoning about a system in physics, the perspective we adopt is that of "input" being the forward advance of time (and sometimes initial conditions) and "output" being the amplitude of the physical quantity(ies) or dimension(s) of the system that interest(s) us. The transfer function, then, is when we switch perspectives to consider the "input" to be a sinusoidal signal (characterized by amplitude and phase over time), and the "output" is the new amplitude and phase of that signal [after "traversing" the system]. Of course, you're actually working with a closed-loop, but most input/output systems can be modeled as a closed-loop if you sufficiently broaden the system's boundaries.

This turns out to be useful for/in several reasons/contexts:

- many physical phenomena are sine waves (or, thanks to Fourier, a sum of sometimes many different sine waves), and often times a system's purpose (to us humans) is to control such a phenomena precisely along the lines of "do this to the amplitude, and/or adjust the phase like so" - dampening, feedback loops, more sophisticated processes like hysteresis, maintaining a steady state given incoming perturbations, etc. In these cases the transfer function ends up being the mathematical expression of that system's function in the "domain language" of that problem, so to speak.

- It turns out that often, when working with systems whose "classical" representation involve components like exponentials or sine and cosine of time (which are "just" complex exponentials of those quantities), the corresponding transfer functions are "simple" fractions of polynomials. More precisely, passing into the Langrange domain allows transforming a differential equation problem into a complex polynomial fractions problem - often much easier to crunch/solve. Furthermore, in the Lagrange domain, de-phasing a signal by pi/2 is equivalent to simply adding 1/(j * signal's frequency) to that signal (if I recall correctly). This makes much of the math more accessible to human intuition, and especially on more complex systems that have several "moving parts" the linear quality of polynomials becomes invaluable.

Personally, I remember quickly adopting, once I'd grokked it, the transfer function perspective when trying to reason about the effect of introducing a capacitor into an existing circuit - analog or DC[0] - as well as things like how the material properties of a door contribute to its behavior as a low-pass filter on sound waves. Sitting down and doing the math, the formulas that I would arrive at spoke much more clearly to me. Also, you are sort of adopting a "time-agnostic" (or perhaps time-invariant) perspective, where the system itself does not change over time. Instead, its' input is characterized by how it behaves over time, and the transfer function (especially when plotted) gives you a clear, direct sense of what the output's "behavior over time" will accordingly be. Notably, it's here that the zeroes of the OP become so meaningful.

[0]: part of what initially started making things "tick" for me was when a professor explained that an impulse on an input signal (i.e. a quasi-instant variation, then back to the preceding "steady state" value of it - i.e. a DC current "turning on"), to a transfer function, "looks like" a sine wave signal with a constant amplitude but monotonously increasing phase offset - again I forget if the rate is constant, polynomial, exponential or what.