Keep Tundra running after using remote start?

This is a good example of two transmitters, that operate on the same radio frequency, but transmit different digital codes. His battery may have had more volts than yours resulting in more RF being generated and interfering with your signal to your truck! There also is the possibility that he was standing closer to your truck than you were, at the same time he was also closer to his truck with you being further away, then his transmitter was captured by your truck receiver thus blocking your FOB from working properly. The second he stopped pressing his FOB button, yours should have worked, assuming you actually pressed the FOB!

Yours worked the next time you tried it, simply because there was no radio signal jamming your FOB receiver in your truck.
Basic RF theory, It's called "Capture Effect" The closer/stronger signal always wins!!!
A key tool in the radio "Jammers" playbook!