We can set up a case of 7 numbers:
$$ 0, 0, 0, 0, 0, 0, 0$$
The average of these 7 numbers is 0. To increase the average to 3, one possible way is:
$$ 3, 3, 3, 3, 3, 3, 3 $$
Therefore, we must increase the sum of the 7 numbers by 21.
The average formula for 7 numbers is:
$$ \text{Mean} = \frac{1}{7} \sum_{i=1}^7 a_i $$
$$ 7\cdot \text{Mean} = \text{Sum} $$
If the mean needs to increase by 3, given 7 numbers:
$$ 7(\text{Mean} + 3) = \text{New Sum} $$
$$ \text{New Sum} - \text{Sum} = 7(\text{Mean}+3)-7\text{Mean} $$
$$ =7\text{Mean}+21 - 7\text{Mean}= \boxed{21} $$