The amendment is long and detailed for a constitutional text, but part of it deals with a tricky question: what to do if a president is critically wounded for a long time like Garfield, or critically ill for months on end like Wilson, or just a loony (a term not used in the amendment). For most of the history of the Republic, muddling through was the approach to presidential incapacity, but in the nuclear age, and especially after John Kennedy was murdered, that didn't seem like such a good idea. Books and movies have made considerable hay about the potential for presidential succession shenanigans under the 25th amendment, often involving a scheming vice president or other official in the line of succession, but in practice those invoking the amendment have been exceeding circumspect.
The amendment also clarified the vice presidency. Originally the Constitution did not, in fact, specify that the vice president becomes president upon a vacancy in the top office. A case could be made for an acting presidency, based on the language of the document. John Tyler did not interpret it that way, however, and the precedent stuck for Fillmore, Andrew Johnson, Arthur, Teddy Roosevelt, Coolidge, Truman and Lyndon Johnson. The 25th amendment explicitly says that the vice president will become president, formalizing the precedent.
There was also the issue of a vacant vice presidency. In between the time that Washington took office and the ratification of the 25th amendment, the vice presidency was vacant for a total of more than 37 years, either because a vice president had become president, or a veep had died in office. The amendment was designed to cut down on the vacancy periods in the number-two spot, and so far it has. In the last 40 years, the office has been vacant only about six months. Had it not been in force in 1974, Speaker Carl Albert of Oklahoma would have become acting president.
Sen. Birch Bayh of Indiana (pictured) is considered the architect of the 25th amendment, though generally speaking the 1960s was one of those periods in which the nation was receptive to changing the Constitution -- as it had been during Reconstruction and during the progressive age of the 1910s. Bayh was chairman of the Senate subcommittee on Constitutional Amendments, and also succeeded in getting the 26th amendment rolling, thereby lowering the voting age to 18. He had less luck with two other constitutional projects, the Equal Rights Amendment and the abolition of the Electoral College.