I'm just completely lost, how does movement and attacking work? And what about battles with people and monsters moving around?
For movement to work, the Players tell the DM where and how they want to move, and the DM sorts out the outcome and tells the Players the result.
For combat, the Players tells the DM what actions they want to take, and the DM sorts out the outcome of their actions and decisions and tells the Players the results.
Rinse and repeat. (It's great fun!)
Reference: Players Handbook, page 6 "How to Play"
- The DM describes the Environment
- The Players describe what they want to do
- The DM narrates the result of the adventure