Consider it as we are choosing $i$ people for a committee, from a total of $n$ people. This can be done in $\displaystyle {n \choose i}$ ways.
After we've chosen that, we would like to choose a super-committee. For each of the $i$ people, we decide whether they stay on the committee or go on to the super-committee (but they are not on both the super-committee and the committee). This can be done in $2^i$ ways.
Summing over all $i$ gives the total number of ways this can be done as: $\displaystyle \sum_{i=0}^n {n \choose i}2^i$.
Now we count this in a different way. For each person this gives 3 options: be on neither the super-committee nor the regular committee, be on the committee, or be on the super-committee.
So we have 3 options per person for a total of $3^n$ outcomes, and so
$\displaystyle \sum_{i=0}^n {n \choose i}2^i = 3^n$,
as desired.