One could say that hydrogen peroxide decomposes because it is unstable. But that would be cheating because this is simply restating the fact that we observe it to decompose in fancier language.
We could use some statistics about the reaction, like stating that the decomposition is thermodynamically favourable (this doesn't explain why a catalyst works but it helps to explain why the reaction happens). But, again, this is just a sophisticated way to repackage the observed fact that the reaction is observed to happen.
However, if we think what is happening at the molecular level, we can get some extra insight.
All molecules above absolute zero have some kinetic energy (that is they are bouncing around and banging into each other with a range of speeds, some slow, some fast). Temperature summarises the average amount of energy present in the compound (but the individual molecules have a range of energies, some higher, some slower than the average).
In a solution of hydrogen peroxide (or in the pure liquid, though I don't recommend ever handling the pure liquid) the individual molecules will be bouncing off each other a lot. Each collision will exchange some energy between the colliding molecules. Sometimes a Collision will involve enough energy that one or more of the O-O or OH bonds will rupture. This will generate some reactive species. Then, sometimes, the reactive species will reform in a way that causes water and oxygen to be formed (the details of the reaction are probably fairly complicated). The products oxygen and water have much stronger bonds than the hydrogen peroxide (or at least eh O-O bond in it) so the probability that they will react to reform hydrogen peroxide is very low.
The probability that peroxide will decompose this way is still pretty low, though, at least in dilute and very pure water solutions. But many substances (including some present in slightly impure water) will catalyse the reaction. The presence of manganese dioxide, for example, provides a different route to the same products (that is pretty much the definition of a catalyst). If a peroxide molecule bumps into a manganese dioxide surface, there is a much higher probability that a reaction will happen than if it hits another peroxide molecule (it probably transfers an electron from the manganese oxidising it and this can be very fast and also can precipitate more reactions that break up the peroxide and, ultimately, yields water, oxygen and leaves manganese dioxide). Human blood contains an enzyme that is designed to do the same job as peroxides and related active species are generated by other biological reactions and you really, really don't want them to build up in the blood. Catalase uses oxidation reactions involving iron atoms to do an even better job of facilitating the reaction than manganese dioxide.
So the detail of the reaction is all about molecules bumping into each other and causing bonds to break and reform in a different way. The overall reaction releases energy (which is to say that the new bonds formed are stronger then the ones that get broken and the reverse reactions are therefore a lot less likely to happen). Catalysts proved an easier route to the products than the reaction where peroxide molecules just bump into each other.