I'm guessing that it became more popular terminology due to two reasons: First, it initially looks like it is less of an insult than "asshole." Often it's used as though someone toxic is literally poisonous and as such shouldn't be allowed to interact with anyone ever, whereas you might tolerate an asshole, so really it isn't any less of an insult. But at first glance it doesn't sound as bad, so you can get away with using it in more contexts. The second reason is related to this, in that it's a lot more vague what "toxic" means when compared to asshole. Someone could theoretically be toxic while not being unpleasant or being a mean person, due to creating a bad environment inadvertently or something. An asshole has to be a bit more active. Thus you can use "toxic" in many situations where you wouldn't get away with "asshole."
At this point it's just been used so much that many people use it as a default without thinking about it.