Normally it's used to get off of the horse's back and let them have more freedom of movement. I much prefer half seat over two point and it just helps the horse really move under you especially whilst jumping. You won't see it often in the dressage ring but you'll see it very commonly in the jumper ring, as well as 2 point and full seat. Two point is used in the Hunter ring. Personally I like to sit down unless the horse specifically needs half seat (I used to had a horse that hated being ridden in a full seat). Hope I explained it well! Good luck!!
You want to close your hip angle. Keep your seat "hovering" over the middle of the saddle. Just like sitting normal have your hips align with your heels. You want to hold your upper body leaning forwards. But not a lot so your laying on the horses neck. Remember keeping your shoulders back and heels down. You may put your hands on either side of the horses neck or grab mane if your still learning.