The scenario everyone worried about, and was beaten into our heads, was a bit of oil or grease leaking into the tires, which is pretty normal on heavy equipment. Under typical conditions, this is harmless. However, under high compression that oil can start to vaporize, which basically turns it into highly flammable fuel. Under the right conditions this fuel when mixed with the pressurized air can and will spontaneously combust. Failures can be pretty spectacular, and they had video to prove it. The nitrogen removes the oxygen from the equation so that the combustion can never occur.

Apparently it happened often enough historically that it is mandatory practice in many domains. For me, this was received wisdom because I've never seen this kind of explosive failure in real life, which is a great thing. But if I put my chemical engineer hat on it also is eminently plausible, so I put it in the category of "lessons they learned the hard way".