On your right panel there is a powerplay button. Click it and you can inspect the individual Powers and pledge to one of them.
There are several activities you can do regarding control, expansion and defense of your power's systems, that generally consist of either killing ships of enemy factions or hauling stuff either from your headquarters to various systems or the other way around (depending on activity and power).
You can get into powerplay at any point, although it's good to say that if you want to have any real effect, it is good to have a big ship and a lot of money and time. You can also do it just recreationally. I do.
If you are in any way serious about it, though, I'd very much recommend googling up a proper PP guide that can explain everything in detail and also consider joining a subreddit of your Power of choice, because by and large, PP is a cooperative activity and it's good to coordinate efforts with others. Otherwise you end up reinforcing a system other people in your Power decided to get rid of, for example, which enrages them.
Also a side note - if you pledge to a power, count on being attacked at random by ships of hostile factions. Even if you play Solo and aren't in danger of running into an opposing player, hostile faction's NPCs can also be annoying and for a newcomer with not the best equipment, even dangerous. You are basically paining a target on your stern.
There are very few actual rewards for participating outside of the good feeling, though. You can earn some change in salary if you don't invest your own money. Every faction offers you a bonus for certain activities. (like extra rewards for bounties, or bonus influence on factions). If you stay with one power for a month, you can also unlock their special module that you can buy as long as you are at least rank 3 in the power. All of this is described in the Powerplay menu on your ship. (Or in any guide)