Edit the Actions and check the run action on 'Select' instead of 'Menu' and check the 'Run on single select only' box.
There are multiple layers to the question you are asking (as I understand it.)
You can pass filter and parameter values in a URL string to a new dashboard or worksheet. If you don't know how to do that, there are a lot of discussions and white papers and examples out there if you google for it.
But the trick I think you are looking for is how to generate the list of values the user has selected in the filter.
You want to check out Jonathan Drummey's example (see sheet 15 in the link) of cycling through your table to grab and build a list of all the selected values.
Go here: Tableau Public
Hmm, I tried that and it's not behaving as I'd expect... here's a screenshot of my edited action... did I miss something? When I change from Menu to Select (the 'run on single select only' was already checked) it seems to remove the linking function entirely. Specifically, clicking "Community Health Action of Staten Island" doesn't do anything, and the menu option as shown in the original screenshot is now gone.