"To engage the player" means to create experiences or interactions that capture and maintain the player's interest, attention, and involvement while playing a game or participating in an activity. Full definition
Depending on the game, action adventure games engage players in experimentation, problem solving, strategic thinking, teamwork, and even creativity. (raisesmartkid.com)
This creates a user experience packed full of Epic Meaning which engages players with the game and its outcome. (growthengineering.co.uk)
Valve has a really good model of engaging players and community in the game through in - game items and contests where people create merchandise. (mmogames.com)