What actually killed them was Volker raising interest rates to 18% in a misguided attempt to cause the inflation triggered by the break up of the Bretton Woods accords and the accompanying oil crisis. (That can also be read the other way round as it happens - one of the several things that stressed the fixed rate currency exchange agreement out of existence was the oil trade imbalance and the accompanying flows of dollars. Volker raised interest rates following economic textbook theory to suppress inflation, and it backfired rather spectacularly.
Unlike today - this wasn't strictly a quantitative money inflation (i.e. nobody had just increased the US money supply by 25%), and this is very clear in the M2/M3 figures of that time.
At any rate. Key thing, then and now - the US residential mortgage market is dominated by long term, fixed rate loans. When interest rates rise very quickly, this creates huge issues for the lenders. The S&L's got caught with a lot of low interest rate, fixed rate loans, and couldn't consequently pay their savers enough to keep their deposits. As savers moved deposits into higher rate institutions it pushed the entire S&L (and many banks as well) into difficulty, some compensated for this by making riskier (higher rate) loans, and the whole sector crashed.
One of the unfortunate side effects of this was that it led to Salmon Brothers developing a massive loan securitisation program (to buy the loans from the S&L as a way out, kind of), and that led directly to the Mortgage Backed Security crises of 2000 and 2006. It also somewhat resulted in the South American crisis in the 1980´s since that was another place the US banks went looking for high interest rate loans.