Questions have popped up a couple of times in the GOTD comments, so maybe this will help someone...
Subs or captioning may be present in different types of video streams. DVD & Blu-ray subs are, like audio, a separate file that's combined with the video. Streaming services, cable, and satalite also use a separate subtitle or captioning stream alongside video and audio streams, though there's no real standard, & they can use their own proprietary format. DVDs may also have captioning embedded in the video itself, but it cannot make it to a TV [so the TV can decode it] over HDMI. Traditional captioning can be included in digital over the air broadcast streams, though like streaming, different formats may be used.
With few exceptions, e.g. some versions of Nero Recode when used with DVDs, subtitles &/or captions need to be treated separately whenever you're not using/viewing the original stream including video, audio, & subs, e.g. with re-encoded or converted video files. There are tools, e.g. ccextractor, to extract embedded captions to a specially formatted .srt text file, and there are tools, e.g. Subtitle Edit, to OCR DVD & Blu-ray graphics-based subs to .srt files. Methods & tools for streaming, cable, & satellite video vary according to the source & the tools you use to download or capture these usually DRMed streams. Methods & tools for digital OTA also vary. Google for whatever video source you're trying to work with.
Once you have an .srt file, the easiest way to view subs is to store that file next to the video file, using the same name for each -- very many players, including OEM players on cell phones, can then display those subs. Video converters can also often import .srt files, but any advantage is dubious -- as a separate file you can turn sub display on/off, and usually have control over how it's displayed, e.g. font, size, shading etc. Note: there are apps or tools to adjust subtitle timing *if* needed, modifying the .srt file. Note 2: graphics-based DVD & Blu-ray subs can be re-used as-is if you've re-encoded the video and are using it in a new DVD or Blu-ray.