Intercooling in multi-stage air compressors primarily serves to: 

Explanation:

Intercooler:

• The intercooler is a constant pressure device which cools the working fluid at constant pressure.
• In a two or multistage air compressor an intercooler is always placed between the low pressure (L.P.) and high pressure (H.P.) cylinder.
• The purpose of an intercooler is to reduce the work input given to the compressor to reach the same pressure as can be reached by a single compressor.
• Here in the below diagram it can be clearly seen that using the intercooler shaded region is the amount of work saved while compression by multistaging.

PV and TS diagram of the compression with intercooling is shown in the figure below:

It can be seen that intercooling is done at constant pressure and is represented by a horizontal line on the PV diagram.

Below shown is the position of intercooler with respect to 1st and 2nd stage compression.