We start with 6x = 8y = 14z. Whenever you have a chance to simplify math by factoring, you should do it, so let's divide each term by 2 to get:
3x = 4y = 7z
Because we know that x, y, and z have to be integers, we're being asked to find common multiples, because we need one number to equate to 3x, 4y, and 7z. That number must be divisible by 3, 4, and 7 in order for this to work - think about it...if x were 2, then 6 = 4y = 7z would mean that y is 3/2 and z is 6/7, and those aren't integers. To get x, y, and z to be integers, then we need a common multiple.
The least common multiple of 3, 4, and 7 is 84, which means that the smallest possible value of 3x, 4y, and 7z is 84. If that were the case, we'd have:
x = 28
y = 21
z = 12
That means that one possible sum of x + y + z = 61. But since we only found the LEAST common multiple of 3, 4, and 7, the answer could be any multiple of 61. Answer choice D is 122 = 61 * 2, so it is a potential answer and therefore correct.
Know that the GMAT loves to test factors and divisibility, so let that be your guide on questions like this. When you see a problem in which variables need to be integers and an equation has to be met, there's a good chance that you're being asked about divisibility, so you can view it through that lens.