It's spinning in such a way that when the lower part of the tether is close to the Earth, then it is moving backwards relative to its orbit, so slower. That lets it be exactly synchronized with the speed of a sub orbital spaceship that can only get up to the lower orbit. Which then hooks onto the tether and gets pulled up into a higher orbit by the system. Meanwhile the tether assembly loses energy so falls into a lower orbit but it can be continually reboosting to compensate e.g. using electrical power generated from solar panels to push electricity through the cable against the tendencies of the Earth's magnetic field, which creates a propulsive force to send it up into a higher orbit again.
Coming back would be the same. Though it's most useful for going up into orbit (saves a lot of fuel that way, while the other way returning from orbit you can do it with no fuel to speak of just parachutes and aeroshell).
If we can build a long enough tether, with strong enough materials, no reason why it can't dip into the atmosphere itself and pluck an airplane instead of a spacecraft into a higher orbit. Have it spinning so fast backwards that it is almost stationary relative to the Earth's surface at the lower point. And if it is a really long tether, then at the lower end for as long as it is in the Earth's atmosphere it would be pretty much stationary. So not a huge amount of friction, and you wouldn't be burnt up as it plucks you out of the atmosphere because you only start to travel at huge speeds when you are well out of the atmosphere. I've seen this suggested somewhere but can't find the reference right now. Not sure if it is possible with present day materials such as carbon fibre etc or it needs future materials.