Why are there distortions in electron cloud in molecules? How does increase surface area amplify these distortions?
The distortions are a quantum effect

Yes, branched chain hydrocarbons tend to have lower boiling points and this depends on their surface area. But the "distortions" in the electron clouds don't depend on the surface area.

The ultimate reason why there are "distortions" (more accurately described as temporary dipoles caused because electrons are not evenly distributed) is because of quantum effects. The uncertainly principle says we can't know the combined location and momentum of quantum things beyond a certain limit. When you work through the mathematics of this, this means that electron clouds always contain random fluctuations that appear as short lived temporary dipoles. The fluctuations in one molecule's cloud can be felt by the electron clouds in other molecules (and can cause second order effects that alter their electron clouds as well as interacting with their random dipoles). When you work out the (fairly complicated) mathematics of this you get a weak, short range, attraction between even non-polar molecules. We call those London Dispersion forces (or Van Der Waals forces).

But the underlying electron fluctuations don't change because of the shape of a molecule. However, the total force of all the fluctuations added up depends on the surface area of the molecule (how can they interact with other molecules except at the surface?) More surface, more opportunity for the fluctuations to interact with other molecules.

So it isn't that the distortions themselves depend on the surface area, it is that in a larger surface there is more opportunity to have them interact and hence the total force caused by the focus is larger.