Add an entry to every element in a nested data structure
You could maybe make your solution a bit neater by using a lambda in place of one of your partials.  assoc might be a little clearer in purpose than a conj as well:

(def sample-board-solved
  (map (partial map #(assoc % :explored true))
       sample-board))


Using map like this will return things in sequences rather than vectors like the input data.  If it's particularly important that you get vectors out you can always use mapv