This is a hollow root (עו, ayin-vav): the middle letter is a vowel, ו, not a hard consonant, so the conjugation behaves differently from a strong root like כ־ת־ב. The core idea is simple and steady: to dwell, to live somewhere. In the bare pa'al you get לָגוּר (to live, to reside); the hitpa'el לְהִתְגּוֹרֵר says the same thing in a more formal register, the verb you meet on official forms. From the same letters Hebrew also builds the word for a resident foreigner, the גֵּר, the one who comes to dwell among you. Every word here is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
Both verbs mean essentially "to live somewhere"; the difference is register. לָגוּר is the everyday word, לְהִתְגּוֹרֵר the bureaucratic one ("place of residence"). Because the root is hollow (the ו in the middle is a vowel), there is no pi'el or hif'il in everyday use; the action is just being in a place, not done to anything, so the causative slots stay empty.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.