Motor not working when connected to motor driver shield?
I cannot tell all of your connections from the photo, but I'm guessing you're supplying power to the Arduino through its power connector and also to the motors, both from the battery. The power connector on the Arduino is specified for 7-12v because the kind of voltage regulator it has needs that 2v difference to operate properly. You're kind of at the edge by supplying 6v, and when the motors come on, their current draw will cause the 6v battery voltage to drop somewhat. It probably drops below the regulator's ability to supply 5v from it, shutting down the Arduino. 

Adding one more cell to your battery will give you 7.5v (5x1.5v) and will may be enough to run the Arduino and the motors. Two more cells would work even better. (Don't make the mistake of trying the small 9v batteries with snap-clips on top - their utterly incapable of supply the current your motors need).