First off you have to get that this take-away is literally a noun (from the word a preceding it).
Oxford says that take-away as a noun means:
A key fact, point, or idea to be remembered, typically one emerging from a discussion or meeting.
Rephrase, the clear take-away (the key fact, the point) meant in the passage above is this:
"It's definitely worth taking note that if you're currently active, you should sustain it. If you're not physically active, you should initiate the habit. This review shows promising evidence that the impact of being active goes far beyond the physical."
Taking an approach means selecting the direction from where you want to start solving a problem.
For instance, I could find myself wanting to teach someone a language.
I could start with just vocabulary, or I could start with grammar.
Those options could be described as a "vocabulary-first approach" or a " grammar-first approach".
Approach comes from the verb approach which means moving closer to a goal. When there are several directions from where to reach that goal, I have the choice of several approaches.
The word is also used in a more literal sense, for instance when an air plane is going to land, it approaches the airport. When it is lined up with the landing strip, and it descends to actually land, we say it is on final approach.
A moment in time can also approach: the end of the month is approaching as I write this.
In your sentence, the word is used in a figurative way. There are supposedly several ways to accomplish what you want to accomplish, and most books take a specific direction to get there.
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