The recommended resoultion (320x240 @ 30FPS) is recommended for performance, not appearance. Real-time image processing is not a simple matter for a computer, and some slower/older machines will have performance issues if you bump this resolution up.
Appearance really does not matter much for shot detection. You can see the targets in HD with your eyes while you are shooting, and as long as the image is legible, you can tell where your shots are in relation to the target.
LASR stretches the image to whatever size your window and computer screen are so that your results are easy to see.
For another example, the first image below is 40x40 pixels.
The second image is still 40x40, but has been told to display at a larger size.
As you can see, making an image larger does not make higher quality. It is still the same number of pixels, but your computer can be told to extrapolate to make it appear larger.
If you are still not understanding how resolution works, try watching this:
You can attempt to bump up your resolution to 640x480 (four times as many pixels) but you may see performance issues (four times as much work for your computer), especially on older/slower machines. This can take the form of lag in shot detection, or taking a long time to lock targets / calibrate a new target zone. If you experience these, you need to take resolution back down.
Other solutions include moving your webcam closer (especially if you are already zooming in), or making any details on your targets more defined (i.e. using a sharpie marker to outline things).
Regardless of resolution, you need 30 FPS (frames per second) minimum!!!