How is it possible for a PMR446 walkie talkie to transmit hundreds of miles away?

You certainly won't be operating through amateur repeaters which are 20MHz plus away from the UHF PMR band or MARS repeaters which are 15MHz+ away. UHF propagation can increase due to atmospherics such as tropospheric scatter which happens mainly in the early Summer months. You'll probably recognise the effect when good ol' analog TV had patterning or one channel on top of another. Tropo scatter can allow UHF transmissions to travel a couple of hundred miles or so. If you are far North enough to have transmitted during an Aurora your transmission range will be extended but your signal will be distorted. Not sure how it sounds on FM but an SSB speech transmission sounds like "a ghost with a sore throat" due to the way the Aurora mangles the signal.