Question
I've heard that mowing can help encourage the grass plant to expand and help build a thicker lawn. Is this true and how does mowing help do this? Thank you.  

Answer
This is correct.

The phenomenon you are asking about is called "tillering". If you cut a grass plant, some grass types tend to send up offsets (side shots) which helps create a bushier plants. In southern part of USA with warm season grasses, mowing new hybrid bermuda grasses at 1.5" will encourage it to spread. Even on cool season grasses (northern half of usa) which should be mowed at 2.5-3.5", mowing regularily produces a thicker lawn.

I recommend mowing twice weekly in spring and fall and weekly in summer (as needed) in the north, and weekly to twice weekly in summer in the south.