The first emote "(｡･ω･)┳┷❤️ ♛ ♚ ♞ ♜ ♟＿(´∀｀●)" is a combination of various chess pieces and a heart, and may be used to express love or enthusiasm for the game of chess.
The second emote "Can I put these pieces on your table?" is likely a request for permission to play a game of chess with someone.