It's not surprising, actually: staying healthy means being both mentally and physically fit, and the two go hand in hand. A person who is truly healthy must exercise both their body and their mind on a regular basis.
As for diet and nutrition, the study supported the notion that eating spinach, broccoli and brussel sprouts is important to good brain health. Having high cholesterol and high blood pressure -- both caused primarily by making poor dietary choices -- greatly increased the risk of Alzheimer's.
Perhaps the most exciting part about this research is the news about brain exercises: those study subjects who engaged their brains in crossword puzzles, reading, social activities or who even playing bingo were at far lower risk of being diagosed with Alzheimer's. Note that watching television wasn't on this list. To be healthy, you have to engage your brain in something new on a daily basis, if possible.
I'm a big proponent of brain exercises, and it's one of the reasons I study comedy improvisation. Some other exercises that are fantastic for your brain include learning a foreign language (the Pimsleur audio CDs are the best I've found), trying to draw or write with the opposite hand that you're used to, brushing your teeth or performing other daily routines with the opposite hand, playing word association games with friends, or playing computer strategy games.