I'm confused as to why ozone is so unstable? I've been told it's because it has a fractional bond order. However, this doesn't make sense to me. Don't fractional bond orders mean resonance and doesn't resonance stabilize ozone?
It's important to bear in mind stability is always a relative judgment. Things are stable or not only with respect to that into which they could turn. So we only say ozone is "unstable" because it can and will easily decompose into oxygen (O2) if given a chance. In some sense it might even be more correct to say what's important here is the stability of O2 (which derives from its strong double bond). If O2 weren't possible, or were significantly higher in energy itself, then we would say O3 is more "stable" just because it wouldn't have anything into which it could as easily turn, so it would stick around longer. So "stability" is not just a property of one molecule, but a property of that molecule plus all the other molecules into which it could turn.
A famous example of this is the high stability of N2 (with its very strong triple bond), which means certain compounds (like TNT or ammonium nitrate) with nitrogen atoms in higher oxidation states (e.g. bonded to O atoms) can readily decompose to products including N2 and O2, producing a lot of energy. That is, they are good explosives, and you can consider that as coming from the strong stability of the O2 and N2 molecules. (This is somewhat akin to the argument in organic chemistry that the pKa of an acid depends strongly on the stability of its conjugate base -- that into which it can turn. The more stable the base, the higher the pKa, i.e. less "stable" the acid.)
All that said, fractional bond order does indeed suggest resonance, because it suggests the failure of a single Lewis structure to describe the molecule accurately. And indeed O3 needs two resonance structures to describe it. It's also correct to say resonance stabilizes a molecule, but, again -- with respect to what? In general, the answer is "with respect to a similar (perhaps hypothetical) molecule that does not have resonance." So O3 is more stable (because of resonance) than some hypothetical O3 molecule trapped in one of the O3 resonance structures. It can still be "unstable" for most purposes, it's just less unstable than this hypothetical non-resonant O3.
A fractional bond order does not mean resonance, it's just a parameter obtained from molecular orbital theory accounting for the difference in population of electrons in bonding and antibonding orbitals. For example, you can use the bond order to explain, in comparison, why oxygen (gas) is more stable than ozone, but you cannot use it to predict ozone stability by itself.