Why is mobility so important? There are many reasons. To me, the most important one is that by checking mobility of pieces, it helps you to see more candidate moves. Seeing more moves helps you to calculate better and I don't know anyone who playes chess that does not want to calculate better.
Another reason is that it when a piece is more mobile, it is a stronger piece. So by checking mobility, this helps you to focus on making your pieces stonger, and reminds you to keep an eye on the more mobile pieces of your opponent.
Mobility can also help you see certain tactics. For example if a piece only has a few places to go, there may be a way to trap that piece. This is a situation that comes up often. It must be looked for either to avoid, or to help you see a move that can often give you a winning advantage.
Just by trying to increase the mobility of your pieces and limit the mobility of you opponents pieces, you can find yourself in better positions with more active pieces. A simple strategy that is always there to fall back on!