What does the propagator in Quantum Mechanics mean? I mean, except from the mathematics behind it, what does it tell us? Is it something that has to do with translations in time?
Propagator in quantum mechanics is just a different name for Green's function for time-dependent Schroedinger equation. It is a unique function that enables us to write, for any time $t_0$,
$$ \psi(x,t) = \int G(x,t;x',t_0)\psi(x',t_0)\,dx' $$ for all $x$ and all $t$. This means the $\psi$ function of $x$ (at any time $t$) can be written as a result of linear operator acting on the $\psi$ function at any other time $t_0$ (usually $t_0<t$ but this is not necessary).
When $\psi(x',t_0)$ is put equal to $\delta(x'-x^*)$ for some $x^*$, we obtain $$ \psi(x,t) = G(x,t;x^*,t_0) $$
so propagator is a result of evolution of $\delta$ distribution governed by the same equation as ordinary $\psi$ functions are - time-dependent Schroedinger equation.
However, $\delta$ distribution is not admissible as $\psi$ in the sense of the Born interpretation, so be cautious when giving $G$ physical interpretation.
The clarity offered by using Feynman diagrams to set up the calculations for a scattering or a decay process also gives an intuitive meaning to the propagator function. The Feynman diagrams are iconal representations with one to one mathematical rules of how to set up, order by order in a perturbative expansion, the integrals which will give a measurable quantity.
Here is a Feynman diagram for estimating the probability of neutron decay , to first order.
The internal lines are represented mathematically by a propagator function which will be integrated over the available phase space.
Virtual particles carry the quantum numbers of the name, in the above case of the W-, but not the mass. Evidently with the mass of the neutron around 1 GeV and the W close to 80GeV the reaction would not go if W were on shell.
So the propagator in this example tells us how the quantum numbers propagate from the initial state to the final state and how the phase space contributes to and kinematically sets bounds to the decay amplitude under study. To be noted, the propagator depresses the value of the integral because of the very much larger mass of W with respect to the energy available for the bounds of the integral .
The same logic holds true for the internal lines for any Feynman diagram, they represent propagators..