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The definition of an estimator "rule that tells how to calculate an estimate " as given here is not clear to me. If I make measurements of some quantity, say age in a group of N people, my outcomes will be $x_1, \dots, x_N$ where $x_i$ represents the age of $i$-th person. One can now talk about the mean as the quantity that gives us "rough idea" about the age of this group.

I want to understand how is the notion of estimator relevant to a situation like above (if at all it is)? What the estimator actually is and how to construct it?

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The mean age would be the estimator, and the mean value would be an estimate, a statistic from the example you mentioned.

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