Looks like it's a bug! Somehow the wheel motors got disabled. By going around to all of them and clicking disable and then enabling them again, it seems to have cleared out whatever bad state was in there.
Short answer : Don't
Rover aren't technically speaking the most time-efficient way to explore a planet. 100km is longest trip i did, and it was insanely long. What's more, as you probably noticed, the risk of accident is still always high even if you are travelling at low speed with a well designed vehicle. I may be wrong, but i doubt anyone sane ever succesfully explored the whole Mun with a rover. That would take dozens of hours.
How the average rover expedition usually ends :
Alternative :
The best way to explore biomes is to use a vehicle that Kerbal Space Program players commonly call a hopper. A hopper is in fact a simple lander with enough fuel to hop around a planet fitted with science collecting devices. If you have a bit of piloting skills, it is a really safer (and faster) option, though it does cost fuel.
Of course, if the target planet has a dense atmosphere allowing jet engine use, such as Laythe, then you'll probably want to use an airplane. For small planets, with low gravity such as the Mun, using a liquid fuel engine powered vehicle is the only option.
Hopping around a planet :
As a reference, here is a wiki article about Hopping around celestial bodies that provides a bit more insight about Hopping vehicles design and strategies.
So why rovers even exists if they are useless for exploration ?
- For the achievement : It is much harder to design a lander for a rover than a common lander.
- For fun : Putting a feet on a planet is something awesome, but driving around with your high tech SUV really makes you the king of this place.
- Sometimes, it can be useful to move things or kerbal around your bases.
- To cover medium-small distances without performing a fuel burn with your lander.
Best Answer
The reason for this is that the default rover control keys (AWSD) are also mapped to vehicle rotation. You don't notice on Kerbin because the rotation power of command pods is too weak to overcome the gravity, but on low-gravity moons the effect becomes much more notable.
There are, however, two ways, to solve this problem.
Method A: Go to the Settings menu and map the rover controls to other keys (many players use IJKL, because that's the keys used by most of the 3rd party Rover mods which existed before 0.19)
Method B: Switch into docking mode, where the AWSD keys control lateral movement, which is only available when you have RCS thrusters on.