First asking why an engine gives you an advantage when you don't understand is a perfectly legitmate question. Now as to the questions:
First some basic opening principles. What you want to do in the opening is develop your pieces and get your king safe (safe king generally means castled with pawns in front of it).
Looking at move 15: all your minor pieces are developed, and your king is safely castled. You also have a pretty strong pawn on d4. Your position is good, but the reason you have a 2 point advantage is your opponents position is really bad.
You opponent has only 1 minor piece out and the queen. Look more closely at your opponents pieces. Neither the bishop nor kinght can move without first moving another piece. Likewise the rook on the king side isn't going anywhere soon. There is no point for your opponent to castle queenside as all the pawns are forward. The king side doesn't look a whole lot better and as previously mentioned it is going to be hard to get the king side bishop and knight out.
Looking at move 28, the perfect world moves for you would have been.
28...Qc1 29.fxe6 Qe3#
The only way for white to defend is to retreat the queen or knight to protect the e3 square on move 29. This give you time to move your light bishop to safety backing up your queen. There is no immediate foreced checkmate, but your queen and light bishop are now partying with your opponent's king while you rook looks down an open file g in that direction too. Also if you took out the pawn on b4 (which you can easily do), you can bring in your dark bishop and several pawns can soon be threating to become extra queens as well.
It's Stockfish 4.
I guess I just have to get many games and analysis under my feet.