One way to achieve that effect is to use data frame clipping.
Here's away to do that:
Have one layer symbolized the way you want for the areas outside the area of interest and another layer symbolized for the part inside the area of interest. Make sure the layer that represents your area of interest or the data for the particular page is on top so it draws above the other layer. These layers can be the same data added to the map twice, with the symbology defined differently for each. You might want to change the name for one or both of the layers in the TOC so you can tell which is which; for example, AOI and non-AOI or Inside and Outside.
Then in the Data Frame Properties, Data Frame tab, in the Clipping section at the bottom, select "clip to current data driven page extent"
Click the Exclude Layers button
In the Select Layers dialog check the checkbox next to the non-AOI layer so it will be excluded from the clipping and will only clip the features in the AOI layer.
Another way is to use a Page Definition Query to only show the non-AOI features in the non-AOI layer.
To do that you would also need 2 layers, one for the AOI and one for the non-AOI. In this case you would place the non-AOI layer higher up in the TOC/drawing order.
Then go to the Layer Proerties for the non-AOI layer, to the Definition Query tab. If everything is set up right (with DDP turned on and the layers set) you should see a button for "Page Definition".
Click that button and check on "Enabled", set the Page Name Field to the field that matches the field you used to define the pages, and select the radio button to show features that "Don't Match".
In either case you could do things like set the layer transparency and layer symbology to get different effects.
Hope this helps get you on the right track 🙂