Why is extreme cold weather a challenge at airports, since jets operate at extreme cold conditions in flight?
The extreme cold is not an issue so much as the weather that typically accompanies it and the fact that planes are not always designed to be started in extreme cold with out some assistance even if they can run in it. For example so long as the weather is good and the aircraft are equipped, there are plenty of airports in Antarctica

When extreme cold is accompanied by precipitation you can get ice build up on the runway which is quite dangerous. Most airports that are in an area of regular cold weather are equipped in some way to handle this. 

Extreme cold temperature that comes with/because of frontal movement may also come with very high winds that can exceed aircraft limits. But high winds can come with warm weather as well so this is not exclusive to the cold.  

For piston engines you have an issue of thick oil. When the temperature drops the oil thickens and it becomes hard to start the engine as well as running the risk of engine damage. Piston engines in commercial operations are much less common than they once were so this is less of an issue. However the general aviation fleet is still pretty effected by this issue. 

Jet engines do have cold start procedures just like piston engines, there is a good discussion on that here. You can see a neat video of an A320 starting up in -28C conditions here so cold weather starts are not a huge issue.

Most airports have some form of ground crew that operate in the elements. Crew like baggage handlers, aircraft marshaller, and even mechanics may work out on the ramp to get an aircraft loaded and ready to go. This is one instance where sheer cold can be a factor although weather can also make working conditions very tough, even if proper protective gear is available.