What does the AdvAgg bundler do? I read the description, but I didn't understand what this component does.
The description says:

If not checked, the bundler will not split up aggregates.

I know the purpose of aggregation is joining files and I take this splitting isn't de-aggregation. What is it?
The description for the CSS bundling option says:

If file count is selected then each bundle will try to have a similar number of original files aggregated inside of it.
If file size is selected then each bundle will try to have a similar file size.

I would like to understand what that means too.
The short answer is to take advantage of the browser cache. It can also be used to take advantage of the number of parallel connections that the browser can make. 

Example:
Front page has a.css and b.css; node 1 has a.css, b.css and c.css. If using one aggregate then the user going from the front page to node 1 would have to download all of the CSS again. By splitting up the aggregate in a smart manner the user will only need to download c.css