From your answer I can only guess which maximal simultaneous traffic you want to be prepared for your page, which is the most important information on how to handle this.
Typically for small websites even basic hosting offers come with enough bandwidth to deliver images and generally also videos. You can lower the bandwidth usage by choosing an appropriate codec and container for your videos. A good resource for that would be Media formats for HTML audio and video | Mozilla. These days it is no necessary to download a whole video before watching it - even without a special video player, using e.g. the HTML5 tags as shown in the Mozilla article, when you choose the right video compression and container the video is streamed. This lowers the requirements for your bandwidth even more.
But if you have need for a more stable, scalable solution for e.g. a highly frequented web site:
For the purpose of delivering bandwidth intense content, often commercial Content Delivery Networks (CDNs) are used. Commercial providers and technology for this are listed and described at Content delivery network | Wikipedia .
A variant is a Reverse Proxy, where for example on this page Reverse Proxy | Cloudflare a provider describes its technology. Because I do not want to promote any provider, I'd like to point to you that Cloudflare is not indisputable, which can be seen from Cloudflare: Criticism and controversies | Wikipedia A reverse proxy is often easier to set up than other load balancers, nevertheless one has to understand the technology to not run into security issues. The technology and for what one has to care are described in this video Seeing Other People's Steam Accounts | Tom Scott. A reverse proxy btw. also helps with avoiding DDOS attacks.
The overall name for this is Load Balancing. Wikipedia has a good article on it: Load balancing (computer) | Wikipedia.
Edit: Btw. if your main interest is overlaying the video with watermarks, I would highly discourage using a custom video player for that, as most browser addons that record video from websites do not capture the screen but save and sometimes even stitch the transmitted video files. A better concept would be to directly include the watermark in the video file.
