Halo 1-3,ODST and kind of Reach are of a different saga, so although important to the whole series, not as important to the current "Reclaimer Saga" which began with Halo 4.
Although it's highly recommended that you play the previous games, especially 4. The Halo Universe is much bigger than just the games, and you can get a much better understanding of the happenings in Halo 5 by reading the books, watching the movies, and other media on Waypoint.
The easiest and shortest method would be to get the Master Chief Collection, as it contains all games that matter, as well as additional lore to help, but if M rated games are banned for you, you will probably be ok just exploring the wikia and the many imgur albums that explain the lore quickly and deeply (just not as fun as actually playing it...).