In essence, the 'assistive devices' feature gives apps the ability to control other apps. It was intended to give assistive applications the ability to read the screen and process information for other methods of interaction, such as VoiceOver or braille readers. However, other apps use the same feature to 'plug in' to other apps.
For example, Steam uses this feature to provide the Steam Overlay on top of games. Apps requiring AppleScript integration also require this feature, as well as apps that want the ability to press buttons in other apps. This would allow assistive apps to press buttons in apps per voice commands or other input, however it could also potentially allow apps to press buttons without your consent.
It doesn't give apps any more access to your data than the app would already have, however there's nothing stopping it from giving itself access to your data by clicking the relevant buttons in the GUI (i.e. opening Get Info and adjusting the file permissions). Despite all this, it would still require that you enter your password.
In conclusion, it simply allows apps to press buttons in other apps and read data from other apps.