Meaning:

Act now without any delay.

If you say "Don't let the grass grow under your feet" to someone, you tell him/her not to waste time and to do something immediately.

If someone doesn't let the grass grow under his/her feet, he/she does something right now without waiting.

Example:

Now that you know what to do, act right away. Don't let the grass grow under your feet.

A: Do you think the manager's already started to do something to solve the problem?
B: I believe so. He's the kind of guy who doesn't let the grass grow under his feet.

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