How does hyperconjugation lead to stability? I know that hyperconjugation leads to delocalization of electrons between filled and unfilled orbitals and thus greater stability (that's what most textbooks said).

However, I think that it reduces the bond strength, because by saying that the electrons delocalized would mean that the electron density between two atoms decreases and hence the 'glue' (bond) that hold the two atoms together becomes weaker. 

Any misconception here? Because my statement seems to contradict with the fact that hyperconjugation increasing stability. Why reduction in bond strength increase stability? (or does it actually decrease the bond strength?)
What you are ignoring is the magnitude of the effect.

While hyperconjugation does decrease the electron density of a localized bond and slightly increases the energy, the overlap with a high-energy, empty orbital is at least very poor. Think that the orbitals are very far from each other and have different symmetries. The total electron density of this bond would remain largely unaffected.

On the other hand, the difference between zero electrons and partial electron density for an empty orbital of high energy is significant, and it increases (to a moderate extent) the stability of, say, a carbocation. Besides, a single orbital often receives electron density by hypercongujation with more than one bond.