Short version:
Rising interest rates make existing bonds less valuable because people can buy new bonds that pay more interest on the same face value.
High interest rates decrease corporate profits (because the companies have to pay more on their own loans). Stocks decline in value both because corporate profits decrease and because when people can earn more from letting their money sit in the bank, they are less interested at putting their money at risk buying stocks.
Long version:
The story you heard about bonds and stocks reacting oppositely to recessions is true. High inflation is a different sort of shock to the system, and the USA hasn't experienced it since the late 70s when many of the parents of people on this message board were still in high school. So people didn't talk as much about, or think as much about, the risk of inflationary environments.
In the long term, stocks recover from inflation. But substantial and sustained inflation can really destroy a bond portfolio.