Well, it's unusual for modern fantasy writers to have chosen to depict this so literally, but the model for this obviously come from mythology. Involving the hero with a triad of women representing maiden-warrior, queen and crone is a motif as old as recorded history. They are not always all his wives (most often, they are not, though I can think of exceptions like some gods and kings in the various celtic cycles for instance, and many examples in even older Indo-European myth).

So the main inspiration for this comes from myth. The motif is not unique in Fantasy (and definitely not in its predecessors, like the Arthurian stories) but often the crone is a mentor, the maiden-warrior or the Queen the love interest and the other the mother, or even an obstacle/foe. Jordan chose to make them all wives of his hero.

As for his motivations for using the motif this way, it's another question. We don't really know, as he's answered questions about this off-handly, joking. But we can guess.

The three women make Rand "universal". Through Elayne, Jordan gets the celtic-Arthurian motif that the King is married to the woman who represents the Land. The Dragon is one with the Land, and through his wounds and his darkness he has harmed it, but through his union with Elayne he also brings it a future (their children, and there's little doubt they'll be male-female. The days were the land is divided by the male-female antagonism are over, Rand and Elayne, both channellers, are going to produce a pair of brother and sister representing the next age).

Then there's Aviendha, who represents among other things Rand's journey as a warrior. She begins the story as one, while he grows into one and becomes a war chief. She is about to turn into a Wise One - and it may be that before he marries the Wise One, Rand had to grow wise too (which the epiphany on DM likely did, while she readies to pass the last threshold to become a WO at the same time). Through Aviendha, Rand is tied to humanity, to his "blood" too. It's through the Aiel that Rand learns his true genetic origins, that make him unique, different from LTT though they share the same soul. Aviendha is also the bridge to Lews Therin and the old covenant between Da'shain and Aes Sedai.

Then there's Min, the "seer", the reader of the Pattern, and now the philosopher trying to puzzle out what Rand needs to do (those helpers are significantly non channellers all.. the channellers have the Power but have failed to come up with a definitive solution for the Bore. It's up to the likes of Fel and Min now, previously Loial also played his part to puzzle out mysteries, while for eg: Moiraine has failed almost everytime she tried to foresee the path Rand had to follow. It's when she surrendered that she got the opportunity to see what she had to do. Min the Seer with her esoteric but passive powers, which comes and go at the Wheels wills, puts Rand - the champion of the Pattern - in the continuum of the past-present-future, the cyclical pattern. She is his advisor in the present, foresees parts of his future and researches clues for him in the wisdom of the past.

That doesn't really explain why Jordan chose to make these three women love interests of Rand. It may well be that he found the idea funny/attracting for many reasons (funny to see the women struggle with that, funny to reverse the gender roles and have women impose this on a prude horrified man and so on). It may well be in part that he found no way that really satisfied him to involve Rand in a significant way with these three characters without making them his lovers. It may also well be that, in a series which thrives on conflicts/shifting character dynamics, he saw there a more interesting opportunity than by multiplying the non-love interests female characters around Rand.

Another "particularity" of the WOT triad is that most often in myth (especially actual myth there, ie:Greek) the relationship of the hero with the triad of women is ambiguous at best when not somewhat antagonistic, with the male being solar in nature, and the women being lunar, and representing the moon phases (rising for the maiden, full for the Queen and declining for the Crone). Jordan has that, but he's divorced it from the love-interests of Rand proper, and split this between two more women, both with a "love" aspect in their relationship with Rand. Lanfear is given the worst/darkest attributes of the moon goddess and the "bad influence" aspect of mythology (bad domineering "mother figure", possessive lover, scorn woman), while Egwene is more ambiguous. Through the story she got a lot of the negative aspects, but there's always the motif present that she isn't truly an antagonist and somewhere somehow the "cosmic union" of sun-moon will happen, only they will co-exist, not form a couple.

Another point that might have influenced Jordan in this is that he clearly sought to make allusions to has many myths, as many versions of myths as possible (often in rather superficial ways, though not really with the triad). That's mostly to support his concept that in his world time is cyclical and our myths come from past Ages and our history will become the future's myths in turn. He wanted this to look as "universal" as possible, with allusions to tons of cultures that often serve, because of a few details here and there, to allude to multiple heroes/gods at once. The thing is, the Crone/Queen/Maiden triad exists not only as a triad, as a whole, but each of these characters also have their own places in other myths. Aviendha/Min/Elayne together allowed Jordan to cover almost the whole spectrum of the various roles of the female mate of the Hero in myth, complemented by characters like Moiraine/Cadsuane/Nyaneve, and Egwene-Lanfear. Jordan would not have gotten that without the triad of wives. This had the effect of multiplying the allusions to world mythology as Jordan sought to. It would not have been so easy without making all three women significant to Rand. He already had Nynaeve/Moiraine/Egwene has his main female non-love interests. Aviendha/Min/Elayne would not have fitted easily in the picture without making them all love interests.

Down the line, the main motivation to go with this, not to have the triad but to make it Rand's love interests, might have a lot to do with Jordan's sense of humour, and his love of naughty stuff. He wrote a all-family sort of story so he didn't push that too far, but he had a bit of fun with that.

As for why Lews Therin didn't have three wives, he really didn't need to. That differentiate Rand from him, making them two distinct characters, and there's also the fact Lews Therin comes from a time where male-female relationships were harmonious, and his "triad" of female characters rather represents not things slowly coming back together, but things falling apart. He's got Mierin the maiden who was using him and whom he rejected, turning her into a dangerous Fury who opened Hell and so on, and become his Crone as Lanfear - seductively beautiful from the outside, but a monstrous creature of the night if her appearance reflected her inner self. He had Ilyena, his Queen, whom he ended up destroying. And as warrior, he had the fellow leader Latra Posae - with whom he developped the central conflict in the Light. As I said, LTT's triad was meant to make the male-female relation fall apart, leaving the world with no "children" and a gender antagonism. Rand's triad is centered on love/friendship (between him and the women, between the women too) and meant to heal things. As a fourth woman, he has Egwene, who is turning into a Mierin who accepts she can't marry the Sun, and together with Rand will end putting back the solar-lunar cycle in order, at some point before this is all over.