How do the engines and the players say if a side is winning or not? What are the key factors they consider while assigning a score to each side?
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There are lots of different things to look at, and there is no hard answer. The best indicator is to look at the amount of pieces on the board. If white have an extra piece over black they are almost always winning. If there is asymmetry in the number of pieces then we usually assign a point value to each piece, pawn being one point, bishops and knights being three points, rooks five points and the queen nine points. So say that white have a queen but black have two rooks then it is likely that black is slightly better then white.
But there are lots of other considerations you have to look at for a full analysis. How much space does each player control on the board, are the pieces developed to useful positions, is the king in a safe position, is there any strong attacks building up, is there a tactic in the position, etc. A computer will analyse all this and adjust the points from counting the pieces to get a score in the same scale. A human will typically not assign a score though and just tell which player is best.
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