It's actually very simple in theory. Walk softly through the woods. Wait for a grouse to flush, or catch it off guard, and shoot it. Eat it. Yum.
But there are some details that you should know:
1. Grouse have excellent camouflage and will use it. They prefer dense undergrowth and shade. They will sit still in it, and you simply can't see them. Sometimes you will see them roosting in trees just above your line of sight. Walk slowly, and scan carefully in all directions. Be silent, and try to become an animal yourself.
2. Grouse don't tend to fly straight up--they fly sideways. If you are in very thick cover, you're not helping yourself.
3. Grouse pair up. You might flush one and miss, but keep your eyes bright: there may be one right behind it.
4. Grouse often come back. If you flush a grouse and miss your opportunity, come back in a little while. You may have success.
5. Grouse love to run. Sometimes what you think may be a squirrel burying nuts for the winter might actually be a grouse running from you.
6. The flush is the key. The grouse's wings make a very distinct sound when flushing (flying away alarmed), and this is sometimes accompanied by a weird "toodle-oodle" sound. Pick up on those sounds; acquire the target; do not analyze, it will be gone; pull the trigger on instinct.
7. If you get a good shot on a grouse and you don't find it, you may not have missed it: keep looking. I recommend doing a process-oriented search of the immediate area, grid style. I once searched for a grouse for a half hour, came back, searched some more, and found it. Again, they have superb camouflage!