When a star first begins the long path toward becoming a red giant, a layer of hydrogen around the core begins to undergo fusion. If this layer was too cold to do fusion throughout the main sequence stage, why is it suddenly warm enough

The contraction in the star due to gravity

Explanation:

The energy of the stars come from the nuclear fusion happening in their core. This reaction generates outward radiation pressure which is balanced by the inward gravitational pull to keep the stars in shape. Very high pressure and temperature is needed for the fusion to start. When all the hydrogen is converted into helium the outward radiation pressure is overpowered by the inward gravitational force.

Because of this core starts contracting and the layer of hydrogen around the core comes in the region where pressure and temperature is optimum for the nuclear fusion to start. Thus nuclear fusion starts in the layer around the core. The main reason is the contraction due to gravity.

this could be newton#'s law of cooling. the cooling curve is exponential