How are the Star Wars Action Battle effects implemented?
In many ways, this subtheme is a new implementation of fairly old idea.

The specialized elements in these sets are the cannons and the arrows that are fired out of them. These elements were first introduced in the Competition subtheme in the late 90s. Here one example set (8233 Blue Thunder vs. The Stinger):



The cannons are a single part that implements a simple spring-loaded firing mechanism triggered by toggling the axle on the back end.



The arrow is a specialized part that fits into the canon and includes a rubberized tip:



While the cannon and arrow elements are fairly non-standard (though by no means rare having appeared in around 100 sets over the years), the other mechanisms involved simply make good use of common System and Technic elements.

For example, here's how the mechanisms that launch the Speeder Bike and Ewok in that set are implemented:



There aren't any particularly special elements involved. The design simply makes clever use of standard parts to transfer the energy from the impact point on the target to other areas of the build causing it to launch various parts off in interesting ways.