Laymans terms so let me give you an example. Let's say you have a web application, and you need to add error logging / auditing. One implementation would be to go into every public method and add your try catch blocks etc...
Well Aspect oriented says hogwash with that, let me inject my method around your method so for example instead of calling YourClass.UpdateModel(), the system might call,
LoggingHandler.CallMethod() this method might then redirect the call to UpdateModel but wraps it in a try catch block to handle logging errors.
Now the trick is that this redirection happens automagically, through configuration or by applying attributes to methods.
This works for as you said cross cutting things which are very common programing elements that exist in every domain, such as: Logging, Auditing, Transaction Mgmt, Authorization.
The idea behind it is to remove all this common plumbing code out of your business / app tier so you can focus on solving the problem not worrying about logging this method call or that method call.