Why is this move Ne5 by white, a mistake ?

Now, If white plays Ne5

Not sure. Question might be, after trading knights on e5, retaking with the bishop, are you any better versus leaving the knight on f3 and instead completing development. You have a form hold of e5, why resolve it now? You could complete development and occupy it later. That might be what the engine is saying. a4 looks interesting, threatening Bb5 pinning and winning the knight, then Ne5. Your opponent would play a6 to prevent it but it gives you some space to manoeuvre on the queenside.

Alternatively Rb1 might be better, attacking the queen, threatening b7, and threatening Bb5 with the idea of exchanging off the knight followed by Ne5. That's a few problems for your opponent to solve.

White needs to play actively otherwise it's pawn down with a bad position. Therefore the only serious plan is to try to play Rb1; hence Nh4 is correct to drive away the strong Bf5 covering b1.

Ne5 is worse than say Be2 because White is actually also behind in development and cannot afford to waste any more time or let the position open with cxd4. The trade of all knights favours Black because Qc1 and Bf4 are actually badly placed in the cxd4 cxd4 configuration.

😔 I'll do a walk of shame.

Thanks Mark, Pfren and Nerwal. I got it. The rook at a1 needs to come out. After Nh4, black decides to open up the centre and captures cxd4. With which pawn, one should capture? The reason why it is e-pawn because it makes sure that you have better control over files. Capturing with c-pawn will result in less control on open b and c ( only 1 pawn in action exerting influence). Or is there anything deep?

The problem with Ne5 is black can play c5xd4 in reply.
White has three moves at that point cxd, exd and NxN
NxN dxc ; Qxc Rc8 (pin to get piece back)
cxd Nb4, which threatens a fork on [c2]
exd NxN ; BxN Rc8, where black likes the look of [c3] or even Qb4+ / Bb4 if white attempts to deal with the backward pawn on the half open file.

Thus IM pfren suggests try Nh4 cxd ; exd Be4 ; when [f3] gets the bishop pair after ... Bg6

It is typical in "London Variations" for black to hit the b-pawn especially if white has gone Colle (?) like with [c3]

edit: Qb4+ / Bb4 from Qb5+ typo error.

Thanks, Rat1960. When i was playing this game, I felt that it was getting complicated beyond my current depth. 

Another suggestion. If black plays f6, why it is a mistake?

Engine changes evaluation from -4.3 to -1.3 there. 

Black should take on e5 then play Qc7 or Qa5 when c3 will fall without any compensation, Black's pieces will be more dominant than ever. f6 is a mistake because now White avoids the trade with Bf4 or Bg3 then the Bf5 and Ng4 becomes quite vulnerable, h3 or/and Nh4 will follow.

#12 If that is white to move the plan would be Bf4/Bd3 and hit black on [h7]
Black wants to hit the backward [c3] pawn, initially with ... Qa5 but if need be additionally with ... Ng4-f6-e4 
If you have played ... f6 you undermine your main queenside attacking threat.
... NxN ?! ; NxN Qc7 (say) ; g4 Bg6 ; when white can ponder [c4] or NxBg6.

Personally I would play ... Qa5 and if Rac1 Ba3
So you have say ... Qa5 ; Bg3 Qxc3 ; QxQ RxQ ; Rfc1 Rfc8 ; RxR RxR ;