Why Bd5 is a mistake?

Bd5 is a mistake; I will explain later, but need some background first.

Game Analysis terminology—the platform assigns 0.40+ for inaccurate moves (loss of tempo, etc), 0.90 for Mistakes (inferior exchanges, tempi loss, etc) and 2.0+ for Blunders (material loss, game loss, etc.)

Game Analysis platform will provide a sequence of moves to support its claim of “mistake, blunder, better move”, etc.

Opening Principles—piece development & tempo; threats take away tempo as opponent cannot develop pieces but must defend or preserve a developed piece.  Moving same piece multiple times loses tempi.

Position Analysis—King Safety (advantage White); Short-term Factors (slight advantage White)—piece mobility as LSB controls long diagonal, quality of position (2v1 developed pieces), material is equal; Long-term Factors (advantage White as Black has isolated pawn, doubled pawn and 3 “islands” vs 2 “islands” for White; space is equal; Tactical Assessment (slight advantage to White—threat on g7; center control is equal.  Summary—White has a slight advantage as noted by the vertical bar in diagram above.

Understanding the above concepts, Bd5 is a mistake as the B will have moved 3 times, losing 2 tempi; allows Nbc3 (developing a piece, opens the back rank  to eventually connect rooks, & threatens d5); BxB with dxe4 opens the d-file & threatens Nd5, a very strong outpost; Black cannot exchange Qs as RxQ will control d-file and trap Black’s King in the center; c6 is forced to prevent Nd5.  White should exchange Qs and castle (control d file with O-O-O+ or Rd1 then O-O).  Summary, c6 is necessary to defend the g2 threat, play it move 8, not 10, losing tempi, position & helping White further its development.  Further, c6 does preserve Black’s option to O-O-O, otherwise, its K remains exposed.  

Sorry for a lengthy post; hope is will be beneficial.  Regards.

In general it is true that you don't want to move the same piece twice, yet that, to me, isn't the big problem of the position. If you look at the position, black has some weaknesses. Most notably the square on f5. Imagine a knight getting there: you'll have no pawn to kick it out. So square f5 is a big weakness (a potential outpost). Yet, it's not just the f5 square, the other light squares are also weak (say e4, g4, h5). Usually when you have a weak color complex, you do not want to trade off the bishop defending those squares. They'll get even weaker. So that's why the trade should be avoided I think.

Now of course it is true that Nbc3 is a good response to Bd5. After Bxe4 I'd be tempted to take back with Nxe4. I don't think black can really play f5 himself, cause those pawns feel really weak. But that's looking very concretely at the position, while I think the weak light squares point to not trading anyway.

Also, black's pawns are all on the dark squares, getting in the way of your dark squared bishop (and not defending those light squares). If you look at the bishop, its only good square is on c5, where it can be active, otherwise it'd be stuck watching its own pawns. Basically your dark squared bishop is your bad bishop, while your lightsquared bishop (not looking at its own pawns) is your good bishop. Trading off your good bishop usually isn't a good idea either. 

And finally there is the fact that black has the bishop pair. The side with the bishop pair usually can have some advantage because of it. That's why strong players dislike trading a bishop when they have the bishop pair and the opponent doesn't. 

I wouldn't be too upset about Bd5 though if I were OP. I would expect this kind of mistake to also be made by high intermediate class players. 

There are many problems with ...Bd5.

1) Why voluntarily give up the Bishop pair in a semi-open position?

2) Black has the inferior pawn structure.  All trading off pieces is going to do is lead you closer to an endgame.  Pawn weaknesses are offset by dynamic factors in the middle game, and stand out like a sore thumb in the endgame.  Black should be looking to trade as many pawns and as few pieces as possible here!  Piece trades favor White here!

Yeah, c6 is better. As ThrillerFan pointed out, there's no need to be swapping of all the pieces since ur kingside is in ruins.

However, I looked at ur most recent game, and u left a bishop hanging for about 5 moves straight. U need to be aware of stuff like this more than subtle positional mistakes like bd5.

But that's not necessarily true. Imagine a rook endgame. How is white going to attack h6 and f6? In fact, shattered kingside pawn structures are quite often no problem whatsoever in rook endgames. These pawns are very different weaknesses to an isolated pawn or doubled pawns on a half-open file. Those can get attacked easily and are big weaknesses that get weaker with the trading of the (minor) pieces. With these pawns I'm much more worried about the middlegame problems of king safety etc. because of the weaknesses on the light squares.

1. Do not move a piece 2 times in the opening. 2. You are giving up you the bishop pair which seem strong in that particular situation. 3. ..f6 restricting your ooponent's bishop, ..Bg7-0-0-Nd7 should be a good developing plan.