How Earth's rotation affect flight times?
It does not affect as much as one would think.

The atmosphere moves along with Earth's rotation. An airplane taking off from one place on Earth has to move through the atmosphere which is moving with Earth.

Now you may ask when does it take longer when we travel from Western Europe to Eastern USA, compared to USA to Europe? That depends on Jet Stream:


  Jet streams are fast flowing, narrow air currents found in the atmosphere of ... Earth.


Many air routes take advantage of jet streams, as you can see as follows:



Pilots know about the jet streams and will try to gain benefit of them. Recently, there were some news that a flight from New York to London traveled faster than expected because it was in jet stream (details here and here).



If you want to know why Earth's atmosphere moves with it, please see this.

There is a similar question on Physics.SE.