Just saying "Trade pieces when ahead" is an oversimplification. Define "ahead". Is a pawn up ahead? The answer to that is often no. Low rated players think material is everything. The most important factor is king safety. Other factors include piece coordination, initiative, pawn structure, the list goes on. It is just like how saying Bishops and Knights being worth 3 and Rooks worth 5 is an oversimplification. If I have 2 knights and you have 2 bishops, is it equal? What about if there are 4 blocked files and very few open diagonals? What about if it is wide open with both sides having a-, b-, g-, and h-pawns?
There is an FM that runs the club I go to, and he does a lecture on Tuesday nights. He will tell you time and time again, that if you are up material, only trade if you are being attacked, like if you won two pawns at the cost of a major initiative for your opponent who has a raging attack, trade things like the queen and Rooks if you can to defuse his attack.
But if you are say, up a pawn, but your real advantage is the initiative, and you are attacking the Black King and he is also behind in development, like his rook is still on a8 and Bishop still on c8, trading down will just slow down your attack, give him time to develop, and maybe enter a pawn-down rook ending and that might actually be drawable.
Even worse, your opponent may have sacrificed a rook for your knight, but with his knight now dominant on d4 (he is black, you are white) and no open files, he may actually be better the exchange down.
Also, you cannot just say trade pieces or don't trade pieces. It often depends on the piece. Let's say you are a pawn up, 7 to 6, and a pair of knights have been traded. It is not as simple as trade down or don't trade down. Maybe his knight is holding the entire position together. Perhaps a trade of knights or bishop for knight causes his whole position to crumble. You want that one trade, but when he offers you a queen trade, you reject it.
Here are the times you want to trade pieces, particularly when up material or even material:
1) You are being attacked. Defusing your Opponent's attack
2) The removal of a key defensive piece for your opponent. This often entails a single piece trade and holding on to the rest.
3) Your advantage is static, not dynamic. A static advantage is a long term advantage. A dynamic advantage is a short term advantage that requires immediate action. For example, take the exchange Ruy Lopez. 1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Bxc6 dxc6 5.O-O f6 6.d4. Often, at some point, dxe5 by White or exd4 by Black is played. This leads to abcefgh for White and abccfgh for Black. Notice Black has 4 pawns to your 3, but can never force a passed pawn. Let's say he plays ...c5, ...c6, and ...a6 at some point while you played a4 and c4. If he now plays ...b5, DO NOT TAKE on b5. Play b3 and let him take you. If ...bxc4, you play bxc4 and you have a4 and c4 vs a6, c5, and c6. 2 pawns easily stopping 3. You have a clean 4-on-3 on the kingside. This is a static advantage that will be around all game. No rush. The more pieces that are traded off, the more his static disadvantage stands out like a sore thumb. Black has the dynamic advantages. More files open for his heavy pieces, the Bishop pair, often the initiative. But unlike pawn structure weaknesses, all of Black's advantages evaporate over time, so Black must act fast and attack, avoiding trades as much as possible.
Another opening where it is a battle of better structure vs the Bishop pair is the Trompowsky Attack. The difference here is doubled f-pawns instead of c-pawns, and White lacks the dark-squared Bishop instead of the light-squared one. Same concept though. In fact, a game I won over the board last night as White in 92 moves started 1.d4 Nf6 2.Bg5 d5 3.Nd2 h6 4.Bxf6 exf6 5.e4 dxe4 6.Nxe4 f5 7.Ng3. Again it is pawn structure vs Bishop pair. Black played way too passively, and static advantages win in the long run, hence the 92-move game, most of it a pawn up minor piece ending - knight and 3 pawns for White vs Bishop and 2 pawns for Black. White had the static advantage, Black had the dynamic advantage of the 2 bishops. Piece trades almost always favor White here.
So as you can see, the blanket statement of trading when up material is a farce. It is about 3 things mostly - Initiative, Development, and Static vs Dynamic Advantages.
I personally prefer to keep pieces on the board and use my extra piece to attack.
I think in the future I will consider this instead of automatically trading down and take it by a case by case basis. Another poster mentioned that trading down is more beneficial in games with shorter time controls, which makes sense.