According to Golf News Net, there are two main reasons why golf tournaments have cuts.
The first reason is that If there weren't cuts, there would always be a lot of players on the course. Many tournaments start out with more than 100 players. So after those two rounds, it's most likely already clear who the top players are and who would win. (The winner is the player with the lowest total score after 72 holes.)
The second reason is that tournaments typically have a specific format they stick to, so players are cut to meet those requirements.
After a player is cut from the Masters, they are welcome to keep watching the games. Although many opt to leave after they are eliminated, the previous winner of the Masters title has to stay. Per The Augusta Chronicle, the defending champion has to help the new title holder put on their green jacket once everything is over.
If you want to watch the Masters, check out the full schedule at CBSSports.com.