Indeed, the "Insert video" action does not do much beyond creating a link. WordPress does not contain a generic player that can play any video hosted anywhere on the internet, but it does support something better: embedding via oEmbed. This means you can put the URL of the video page in your content, and it will replace it with the proper embed code - if the video site supports oEmbed (most popular sites do), and if WordPress is configured to allow embed from this site (but you can enable oEmbed autodiscovery or use a plugin like Embedly to add more sites).
The support in the editor could be better, but this is probably the most future-proof way to embed videos in your posts.