The Intel 8080 microprocessor is very powerful but it has limitations. The shift function on the acumulator has a binary equivalent of multiply by 2 or divide by 2.
A barrel shifter is simply a multiplexer with however many bits need to be shited. The 74LS157 is a quad multiplexor in a 16-pin DIP.
This can make it a lot easier to move the game elements in the display buffer. This relieves the 8080 to do other things.
The game Pong used barrel shifters to move the paddles back and forth. Other barrel shifters handled the movement of the ball.
The TTL logic in the image is simply several 2 line units. So six 74LS157 will be needed for the 8-bit version shown. A 4-bit version uses fewer 74LS157 chips.