Recall that the formula for choosing k people from a group of n people is
`((n),(k)) = (n!) / (k!(n-k)!)`
The answer to your question follows directly from this formula. In both cases, the n=12. In the first case, k=4, so the denominator is 4!*8!. In the second case, k=8, so the denominator is 8!*4!.
This explains why it's true by using the formula, but let's try to develop a more intuitive understanding of why this is true.
Suppose I choose a 4 people from a group of twelve. Since I chose 4 people, this also means that there are 8 people that I did not choose. So in choosing 4 people, I am simultaneously NOT choosing 8 people. So the number of ways I can choose 4 people from 12 is the same as the number of ways I can choose 8 from 12.