Why does higher inflation translate to higher rates?

Answer 1

It's part of the interest rate. There is a premium for inflation.
Inflation means that today's money is worth less than yesterday's money. You are paying the money received today with tomorrow's dollars. The banks want to protect themselves against inflation so they charge higher rates to compensate for it.

Answer 2

Because lenders want to recapture real losses they experience because of inflation. So they factor their inflation expectations into the offered interest rates.