Atmospheric pressure decreases with increase in altitude. Therefore, as we go higher the pressure within the blood vessels exceeds the outside pressure. This causes the blood vessels to rupture. As a result, a mountaineer suffer nose bleeding, ear bleeding etc.
At high altitudes, there's less oxygen available. This makes the air thinner and drier. The climate inside your nose also gets drier because of this. This will also makes the blood vessels exposed inside your nose more likely to burst there than at lower altitudes.