TCU's invite to the Rose Bowl is the last part of a story that began in 1935. The Bowl whittled down one of its choices for a bid to the winner of the Game of the Century, TCU's season finale versus SMU. The Mustang's Bobby Wilson made one more play than Horned Frog legend Sammy Baugh, and the boys from Dallas won the trip to L.A.
Here's Richard Durrett's recap, and a highlight video showing one of Sam Baugh's slinging passes (incomplete) at the end:
TCU wasn't out of luck, though-- they sent to New Orleans and won the Sugar Bowl, and the national championship. They repeated the feat three years later, having beaten SMU in that year's season finale, Davey O'Brien at the helm.
But the Rose Bowl hasn't approached TCU since, making this year's bowl especially fulfilling.