Two words: BABY BOOM
It began with a sudden surge of births in 1946 and ended with a long decline in births in the 1960's.
There is a certain age range when young men are statistically prone to committing crimes than other ages, usually from around 16 through their mid 20's.
Take a look at that chart again with that in mind - the first wave of 1946 Baby Boomers would have turned 16 in 1962, right where the murder rate begins to rise. The last of them, born in the early to mid 1960's, would have turned 25 in the late '80s and early 90's, when the murder rate began to fall.
Correlation does not imply causation in either case, of course, but there's more to this issue than the simplistic picture that this chart paints. Some people are claiming the murder surge occurred because of leaded gasoline.