As already outlined by the others, the "Max-Age" headers influences caches more downstream from AEM, that means any well-configured intermediary, including CDNs, and also your browser should respect them.
Regarding dispatcher: In most cases it must not affect dispatcher at all. If the Max-Age has passed and clients reach back to your dispatcher, the dispatcher might deliver the same file again (with a new Max-Age header) if that file has not changed in the cache. Which is default if you invalidate your dispatcher cache on any change.
The situation is a bit different when you use the feature of TTL-based caching on the dispatcher. Because then your application delivers content, and you cannot invalidate the cache properly (typically because the invalidation logic would be too complex/demaning). In that case you enable that feature and let the dispatcher invalidate the files after a few minutes automatically, and refetches the files again from AEM.
Related to browser max-age a new request will not be made to the server as the js/css/image is stored in the browser cache (client side). After the max-age it will try to get the response by hitting the Akamai/Dispatcher/Publisher. If the response is cached in Akamai/Dispatcher it will not go to publisher.