How does hyperconjugation explain the stability of alkenes?
As always in chemistry one phenomenon (experimental observation) can be explained with different models, that live in different "physical model worlds", and more often than not these models are not consistent among themselves and might be even bare of a physical reality, especially if they are analyzed in rigorous way, nevertheless might be very useful tools, that take the justification of existence out of the fact that they work in many cases. 

That said,  especially the concept of hyperconjugation itself can be employed at different levels. A very primitive model, but one that works astoundingly well, is to assign a positive inductive (electron releasing effect) to alkyl chains that is increasing with the degree of substitution of the ipso atom. That leads to the idea that this (however small) charge that is pushed to the sp$^2$ carbon atom stabilizes its slight positive polarization (since the majority of electrons in the double bond is above and below the plane of the atoms engaged attached to the doubly bound atoms. The physical reality of this model if often drawn into doubt, which is good to know, but still its simple and powerful and thus there is no reason to abandon it.

At a higher level of model, using a MO picture the idea is that the (empty, anti-bonding) $\sigma^*$ orbitals that come inevitably with the C-C and C-H $\sigma$ bonds can act as kind of "acceptors" for the $\pi$-electrons of the double bonds if in the right orientation. It is important to note that in this idea to actual overlap that mediates the interaction is thought to be established with the part of $\sigma^*$ orbital that is opposing the bound atom. That donor-acceptor interaction ultimately leads to a stabilization of the system. Again the physical reality is also here at least disputed, but however less than in the former case. But still both have in common that the Substituent acts as the donor and that this causes stabilization.

So yes in this higher level of description of the hyperconjugation the reason for the stability is the increasing number of possibilities to form stabilizing overlap.