They were more compact than alternatives at the time, and had lower ESR. This made them especially desirable for supply decoupling in digital logic, whose rando

They were more compact than alternatives at the time, and had lower ESR. This made them especially desirable for supply decoupling in digital logic, whose random fast edges means a lot of broadband noise to shunt — you can't just design to a particular frequency range like you can with most analog circuitry. Digital was also the new expensive hotness, and putting in a tant wasn't going to be a big fraction of BOM cost. These days my understanding is that ceramics and non-tantalum electrolytics have gotten good enough to replace most tantalum uses, and between that, the expense, the slavery, the exploding, and the environmental damage, they're less common than they once were.