Primarily used to grease the leather gasket in the pump, to keep it flexible/pliable so that it can contract and let air pass on the up-stroke, and expand and trap the air on the down-stroke.
The Silicone Grease can also be used on o-rings to keep them from drying out and cracking, as it acts like a lubricant to reduce friction when an o-ring slides into place or rotates.