There's no such thing as a "paid Roku subscription". Roku offers subscriptions to premium services that are provided by 3rd parties, but there's no "Roku subscription".
Never having seen "The Roku Channel" on anything other than a Roku device I'm only going to be able to take a guess. Hopefully there's someone out there that has some experience, but this isn't a "Getting Started" issue which is really only about getting a Roku device up and running. However, I don't really see a place in this "Community" for such a question.
Not having a Roku device shouldn't have anything to do with how the app runs on a Samsung TV. When using "The Roku Channel" on a Roku device you press the asterisk button on the remote while watching a video to bring up settings like captions. I don't know if there's an equivalent button on your Samsung remote that you could try. Sometimes caption settings are buried within the settings for a particular app, so you might want to hunt around within the Roku app to see if you can find anything.
Good luck! If no one else offers a suggestion and you figure it out on your own post back so that others may benefit.
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