As someone who plays an Andorian First Officer in another game... Well, honestly, it really doesn't come up much. *shrug* Mostly because her spouses and children aren't on the ship, so they're basically just background details.
But the best explanation I ran across more or less comes out as this: Think of it like Kangaroo reproduction. Kangaroos give birth very early on in development compared to humans, but the newborn then relocates into the pouch to continue developing. Except in Andorians, the pouch and the ovaries belong to separate genders (the zhen has the pouch and then shen has the overies).
As for the two male genders, the given explanation is that Andorian DNA is structured with chromosomes that bind together in triplets (i.e. a Triple-Helix) instead of pairs (like a human's Double-Helix). One chromosome each comes from the thaan, the chan, and the ovum of the shen. Going by the descriptions given of the process (that reproduction needs to happen in a specific order, thaan/shen then chan/shen) I assume that the thaan's chromosomes bind to those of the chan and shen on specific sides (and the chan and shen's chromosomes cannot bind directly to each other without a thaan's in between), which is why you need one of each male gender.
Net result, three genders required to produce a baby, and a fourth required to gestate it.