BattleForge is capable of saving replays of the games you play. You can either enable the option to save all replays from the options dialog, or you can save replays from the statistics screen after a game.
Replays are stored on your hard drive as a *.pmv file. They can be found in the "My Documents\BattleForge\replays" folder. When downloading a replay this is also the folder where you must put the replay in order to be able to watch it.
Once in the game you have to open the Chronicle to play the replays. The replay tab is on the first page and it is the bottom tab on the right.