Relax. Take some deep breaths and let the tension drain out of your muscles. Frustration constricts your movements. This isn't a skill that can be learned in an instant, so be patient.
Warm up with some stretches and a few slow practice throws. That's right-stretches. Card throwing isn't the most strenuous activity but getting one to fly 50 feet requires a quick, whip-like action. Any athlete who does that sort of movement with cold muscles is just asking for an injury. So do a few fist clenches and rotate your wrists. It'll help loosen up those joints.
Visualize your throws. No matter your chosen technique, card throwing requires timing and muscle memory. In order to get a nice fluid motion, there are a lot of joints that need to work in almost perfect accord. Eventually, with practice, this will happen without even thinking about it. Picture yourself throwing in slow motion, and watch your wrist and fingers unwind in a smooth, continuous movement.
Remember that you're not swinging a baseball bat here. Your arm's full range of motion shouldn't be used. The card needs a high rate of spin, which is accomplished mostly in the wrist. The larger arm muscles just get in the way, although seasoned card-throwers can use their elbow to more fully extend their arms, thus adding to their range.
Practice, but remember to keep it fun. Unless you're planning on somehow making a living throwing random objects, this isn't something that should keep you up at night.