For a complex number a + bi, the conjugate is a - bi. OK let's choose two complex numbers, 2 + 3i and 5 - 2i and unravel this mystery.
1) The "conjugate of the sum" means to add the two complex numbers, then take the conjugate of the sum, as follows:
2 + 3i
+ (5 - 2i)
7 + i
The sum is 7 + i. The "conjugate of the sum" is is 7 - i.
2) The "sum of the conjugates" means take the conjugates of the two numbers we started with, then add them.
The conjugate of 2 + 3i is 2 - 3i. The conjugate of 5 - 2i is 5 + 2i. Now let's sum the conjugates:
2 - 3i
+ (5 + 2i)
7 - i
Which is the same answer we got in part 1 above. Hence the "conjugate of the sum" equals the "sum of the conjugates". Now you too can baffle your friends with your knowledge of complex numbers!