On your go, you can either bolster your attack score, move pieces, produce stuff, or buy in resources. Each player has their own control board, and you indicate your choice by putting your pawn on the relevant square. Once you've done the action, you can choose to do the other actions on that square, which upgrades a power (produce more, move faster, increase your popularity, build permanent structures) for a cost. Get four steels and you can release one of your four mechs onto the board. You can't, however, do the same action on your next go.
Only the mechs and your large player character piece can actually enter opponents' hexes and fight. If you win the mechacombat, the opposing mechs flee back to base along with any little people, but you lose one popularity point for each little person who flees from your almighty mechanical megakilling tractor of doom (yes, the yellow player has mechatractors).
Each player has six stars, and when certain things are accomplished (eg getting all little people out, winning in combat, building all the four structures, getting max attack or max popularity points, achieving an objective), they get a star against that entry on a tracker. When someone gets six stars, the game ends. Depending on how popular you are, you get bonuses for how much land you hold, how many resources you have, and how well-loved you are. Person with the most money (doubling as victory points) at the end wins. So you don't need to fight if you don't want to and yet still win just by being excellent to people. But that's not as fun as unleashing giant mechs on unsuspecting yokels.