Why would this rearrangement happen?
Is methanol even a strong enough reagent for such a reaction?


Methanol is a very poor nucleophile, so the rate of the $\mathrm{S_N2}$ pathway will be extremely slow. This is much likelier to be a $\mathrm{S_N1}$ reaction. The starting material is a benzylic bromide from the point of view of the alkyl halide, both pathways are quite possible, but the lack of a good nucleophile suggests the predominance of the $\mathrm{S_N1}$ route.

So, you go through a carbocation intermediate, which undergoes an alkyl shift to generate a more stable carbocation.



Both carbocations are benzylic and secondary, so hyperconjugation isn't the driving force for the rearrangement, as your comment hints at. Instead, it's simply related to the relief of ring strain as you expand a four-membered ring to a five-membered ring. Four-membered rings are pretty unstable to begin with and fusing one to a benzene ring increases the strain even further since you impose more geometrical constraints on the ring.

Interception by methanol and deprotonation gives the desired product.