Connecting AND Gate Chip to an Integrated Circuit
You connected the DIP switches between Vcc and the AND gate's input, and that's wrong. A floating TTL input (DIP switches off) is seen as logical 1, and when you close the switch you just enforce that 1. So inputs are always seen a 1, and output will be 1, and the LED will light. 

Two things:  


connect the DIP switches between the inputs and ground  
connect the LED between output and Vcc. The logic will be inverted, but the output can sink more current than it can source, and your LED will light more visibly. Check the LED's polarity: the anode goes to Vcc. You also have to add a 150 \$\Omega\$ resistor in series with the LED to limit it's current.