The air we use to inflate our tyres (and breathe) contains mainly nitrogen (78%) and oxygen (21%). Whilst this is fine for general use there can be some benefits in using only nitrogen:

A) PRESSURE LOSS
A tyre's rubber sidewall is not totally impermeable to air, and minute pressure loss over time is normal. Nitrogen actually diffuses through a tyre's rubber sidewall 30-40% slower than Oxygen, however this benefit is very small and a regular pressure check is still required.

B) PRESSURE INCREASE
Moisture in a tyre's air chamber causes greater pressure build-up as a tyre increases in temperature. Nitrogen does not support moisture and behaves as a 'dry' gas. This has no effect on normal tyre use, but in the high temperatures generated through competition (typically 90-100°C) means the difference between 10psi and 6 psi build-up. This change in pressure results in changed handling properties during a race.

CONCLUSION
The benefit in use of Nitrogen for normal road use is confined to a slight reduction in air loss over time. For competition use however, Nitrogen minimises pressure changes and helps to stabilise tyre performance.