Mechanics as narrative is essentially the mechanics and systems within a game communicates to the player either a theme, idea, or narrative. This is a good design snack as it creates a cohesive experience between the gameplay and themes. There isn't "ludo-narrative dissonance", so to speak.
One of the games that does this best is Starbreeze Studios' Brothers: A Tale of Two Sons. The player controls both players on one controller, each inhabiting a joystick. As the player goes through the game, the brothers have to do different things at the same time, forcing he player to use both of his hands as one unit, paralleling the bond between the brothers. Whenever one of the brothers is injured or missing in action, the game feels much different as only one hand is being used.
The metaphor is that each hand or each half of the controller represents a brother, therefore enhancing the narrative experience.
Chariot is a game where you play as a princess and her fiance carrying the King's coffin to its final resting place. Like Brothers this game ties the narrative into the mechanic, throughout the game the player(s) have to lug the coffin around multiple levels.
Mechanics include pushing/pulling the coffin, riding the coffin and sometimes using the coffin as a platform to get to a tough to reach spot. Chariot is an excellent platformer in this sense, it forces you to think about how to not only get your character to the next platform but also the coffin and how it can be used in achieving this goal.
Assassin's Creed IV: Black Flag Edit
Playing as Aveline in Assassin's Creed IV you are given the choice of 3 specific costumes (slave, lady & assassin) which act as more than just cosmetic changes. They provide a series of speed and weapon benefits but more interestingly they change how you are treated walking around the game world. As a slave Aveline is mostly left alone by thugs and can easily blend into the crowd. Your primary focus dressed as a slave is how the slave owning NPC's see you as a slave. Are you working? Are you misbehaving? However dressed as a lady Aveline is beautiful and becomes a target for thugs. She now isnt 'just a face in the crowd' and needs to change the way she acts because of such. In this game, Ubisoft has turned a simple costume mechanic into a metaphor about the constant pressure women and slaves are under to conform to expectations about who they are and what they should be doing.