The air pressure is really a measure of how much the air above you weighs. As you ascend, you remove the air you climb through from the weight felt by the altimeter.
If the air you climb through is cold, and hence has higher density, you remove weight at a faster rate and the altimeter will indicate a faster climb for the same actual rate of climb.
Think of it as the atmosphere having a constant weight, meaning the more the air below you at a given altitude weighs, the less the air above you will weigh. Denser (colder) air below you -> higher weight of the air below you -> lower weight of the air above you -> lower pressure -> altimeter overreads -> look out below!
Cheers,
/Fred