In Japanese, there is actually an onomatopoeia/mimetic for the act of staring: じー jii (or something similar to that - mimetics don't generally have strictly prescribed spellings as far as I'm aware). To the English speaker, this seems bizarre, because staring is not an action that produces sound, so why would there be an onomatopoeia for it? Nonetheless, for whatever reason, Japanese is a language rich in mimetics, and often has mimetics for concepts that lack equivalent onomatopoeia in English.
So to the Japanese audience, it isn't terribly weird that there would be a "sound effect" accompanying someone staring at someone else. Now, I don't know that it's necessarily often the case that "the character actually says 'stare'" - certainly the character's voice actor will say jii, but I don't think that this corresponds to the character themself saying jii. I think the idea is more that using jii there as sort of a "sound effect" just emphasizes the fact that someone is staring at someone else.
Anyway, this causes a problem for subtitlers - clearly something is being spoken, but this particular utterance is something that simply does not and cannot occur in English. I don't know what a good solution to this is, and I guess that nobody else does, either, and this is how we end up with subtitles that make it look like the character is literally saying the word "stare". Which looks really stupid in English, but is there a better way to go about it? Not that I'm aware of.