In blackjack the object of the game is to get as close to 21 with out going over. You're dealt two cards and asked if you want to hit (get another card) or stay. The sum of your cards is across the top of the table and the amount of cards you have is down the left side. Using this table answer the dealer with either hit or stay and presumably win the game. The dealer follows the same rules as this table (or similar) even if they already have you beat.
Casinos will give you this table to play with while you are there because other players at the table will throw a fit if you don't hit/stay at the appropriate times and it ends up costing them a win.
I think the number in the left column is the lower card in your hand. The sum of your two cards is along the top. Use the grid to know whether to hit or stay.
Look up black jack basic strategy. It's actually like 3 times that chart. The numbers along the top are what you have, the left column is what the dealer has showing
Casinos will give you this table to play with while you are there because other players at the table will throw a fit if you don't hit/stay at the appropriate times and it ends up costing them a win.