Question

How does a foam-filled winter coat help keep a person warm?

It slows down the transfer of thermal energy from outside to inside the coat.
It slows down the transfer of thermal energy from inside to outside the coat.

Answer 1

Primarily the second: it slows the transfer of thermal energy from inside to outside the coat.

A foam-filled coat traps many tiny pockets of air, and air is a poor conductor of heat. That greatly reduces conduction (and limits small-scale convection) so body heat leaves much more slowly. (In general an insulator slows heat flow in both directions, but for a winter coat the goal is to keep the warmer body heat from escaping to the colder outside.)