It would be hard to give you rules that work in every position, but here are a few tips that suggest h5 is good here:
1)
Black's main pawn chain "points" to the kingside, so it's natural to seek play there. This is a natural way of playing because you'll have more space to maneuver your pieces than white.
2)
The center is closed and can't be opened quickly, so you don't have to worry about development as much and can afford some pawns moves. If white could play a quick c4 and or f3 and force you to defend then you probably shouldn't waste time on h5.
3)
Your rook is still on h8 and the pawn on g3 gives you a "leaver" or in other words h4 is a pawn break. If white plays h4 to stop you from playing h4 yourself, then you permanently control g4 and can put a piece there. White will not have the g5 square for a knight/bishop because of the awkward development of white's minor pieces.
With black to move the engine recommends h5, which is a move that I would never play and find hard to explain. I guess you take some space on the king side, but isn't it far more important at this point to just finish your development? Can anyone formulate a chess rule from that justifies h5?