Time

Can someone explain how come the person I am playing gets time added to their time after each move?

One of two things:

1) You play a lot of games with an increment. In your history I see a lot of games played under the 15|10 time control. That's 15 minutes with 10 seconds per move alloted.

2) Lag. Chess.com accounts for the first 2 seconds(2000 ms) of lag and refunds that time to the player. If their ping is greater than this, they simple lose the extra time taken in transmission of moves.

Chess.com is riddled by time and connection problems. There is no way lag accounts for the clock stopping so often and for so much. Chess.com is a good site as a chess resource and casual practice but no good if you want a gauranteed fair game.