Brief Summary
The Set-Up
A bunch of rich dudes with swords got mad at their king for the super annoying way that he did whatever the heck he wanted to all the time. After a tense meeting in a soggy meadow, they all agreed to a plan that would limit the king's power.
The Text
First—and most importantly for the people who wrote it—the Magna Carta was designed to get the barons and the king to leave that soggy meadow without slicing each other up with their big, sharp swords. It contains several clauses about the release of prisoners and about everybody going home happy and with all their limbs intact.
Second, the Magna Carta was a list of grievances that could easily have been titled, "Things We Hate About King John." It's a long list, but most of the complaints come down to the fact that King John was abusing his ability to make up new taxes, claiming pieces of the English countryside for himself, and punishing people without letting them stand trial. A lot of the clauses are King John promising not to pull those sorts of shenanigans anymore in every detailed way that the barons can think of.
Finally, the Magna Carta ends with some bonus clauses clearly written by an overachieving baron angling for some extra credit. It specifies that to enforce all the other clauses there needs to be a council of 25 barons to basically keep King John (who they still don't trust) in line.
TL;DR
An unpopular king accidentally agrees to forever improve the world's governments.