As mentioned, HDTV's achieve this effect as described by using Motion Interpolation to generate intermediate frames, reducing the choppiness of the movement. If you are streaming the show to your TV, ensure you have such a feature (Motion Plus, TrimensionDNM, Motionflow, etc.) on and you should notice an increase in frame rate.
In case you're wondering if there is really a difference, here is a video showing 24 FPS and 60 FPS anime side-by-side. Ensure that you set the YouTube player to to 1080/60FPS to be able to notice the difference.
This video highlights Motion Interpolation generated by Smooth Video Project on a Windows computer. The free version of their software will allow us to increase the frame rate of our shows from 8/12/24 to 60FPS.
It is important to note in general, anime production companies do a good job at creating smooth action scenes without the need of more frames. The cases that SVP is useful is generally limited to computer generated imagery. Anime that consists of mainly CGI and 3D visuals benefit from this interpolation especially well in cases where the background imagery is rendered in 2D, as it will make the character movements appear much more fluid. While maybe anecdotal, I personally tested it on Sidonia no Kishi and feel like it did a nice job to resolve the choppy animation in that show.
For information on setting this up, refer to their manual. SVP requires the installation of additional software, ffdshow and LAVFilters in order to process video. SVP is not a video player, this page has a list of compatible video players.
Additional reading on FPS in anime: What does it mean for animation to be done “on ones” or “on twos”?