How does temperature and other factors affect sigmatropic rearrangements? The image below shows a [3,3] sigmatropic rearrangement. My question is why does the increased temperature favour the right side? The arrangement is technically the same except for the methyl groups on different carbons.

Here is another example:
The increased temperature is necessary to overcome the activation energy of the reaction. Many [3,3] sigmatropic rearrangements have very high activation energies. The products are favored at equilibrium because they are thermodynamically more stable. For example, in the first case, you are exchanging monosubstituted alkenes for disubstituted alkenes.