As dartme1 writes, this appears in Biblical Hebrew and in 100+ years old literary Hebrew. Doesn't work in spoken Hebrew in present and in future tenses; you do need the pronoun. Strangely, the pronoun is omitted in the past: "למה בהית בי?" (Also in declarative, BTW: "בהית בי."), and added only to stress - "למה אתה בהית בי" would be used only if stressing אתה, to mean "Why did you, of all people, stare at me?".
It can be understood why the pronoun is required in present tense - the verb form is much less specific. I'm not sure why the difference between past and future, both are about as specific.
I only know ancient Hebrew. Ancient Hebrew regularly uses inflected verbs and no pronouns. So far in Duolingo, I don't think I've seen any perfect or imperfect inflection. What you wrote might be translated: Why is [there] one staring at me? (though this shoresh wasn't used back then to my knowledge.) lamah יִבְהֶה bi? is "Why will he stare at me?" lamah בָּהָה bi? is "Why was he staring at me?" and different verbs for feminine.
Taken from https://www.pealim.com/dict/181-livhot/
Biblical Hebrew is a very ancient language, people thought differently in those days. There was no present tense verbal system, as such, so the participle was used to fill in the gap. But the participle also had other uses, especially to designate people who normally do a particular activity. The participle was also used for continuous activity in the past. As it wasn't an active verb, personal pronouns were often added. In other circumstances, when verbal forms were used, those forms already included a reference to the subject of the verb, so personal pronouns were not used. In fact they were often not used for objects of verbs either, as they were tacked onto the end of the verb. When personal pronouns were used, it was either for emphasis, or to clear up ambiguity that sometimes existed when the subject and object of a verb were the same number and gender. This was potentially a problem in Biblical Hebrew because the verb was the first word in the sentence, then the subject, then the object of the verb...It's a bit like having the sentence "A bird, she ate", in English. When Hebrew was revived and made into a modern language, it was given a European word order and the participle was designated to be used as a present tense verb, with personal pronouns. A few Aramaic words were thrown into the mix, after centuries of close association with Hebrew. Most grammatical forms were retained, in deference to the holiness of the language, even if some of them are not used very much. It's normal for modern speakers to want to use other modern languages as reference points when talking about Hebrew, but in fact if you understand the past it is much easier. Languages are always in a process of change, every native speaker has to decide for him or herself where they stand in terms of upholding tradition or embracing new forms. For us, it is enough just to understand.