There are several ways to do this. Try one of the following.
At count down tap Right Trigger and Left Trigger then HOLD Right Trigger
Now your back wheels should be spinning and smoking. If they're not repeat quickly. Make sure you get them spinning before the count down goes from 2 -> 1.
3, 2, 1 ... GO. On go tap Left Trigger, Right Trigger and A.
Same as above but on GO tap Left Trigger and press Right Trigger and A.
Same as 1 but Just press A on GO. This is the hardest way to do it since it requires perfect timing.
3, 2, 1... GO. On go tap Left Trigger and press Right Trigger. This is also and advanced technique and requires perfect timing.