The starting point is the most concrete thing in a dry land: גֶּשֶׁם, rain. From rain, water that becomes physical, that touches the ground, Hebrew built the idea of the material (גַּשְׁמִי, corporeal) and then, beautifully, the idea of making a thing real: to fulfil a dream, to implement a plan (לְהַגְשִׁים), and for that dream to materialize on its own (לְהִתְגַּשֵּׁם). One root reaches from the weather report to the deepest word for realizing your vision. 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.
The verbal life of this root is the hif'il / hitpa'el pair, both abstract and both built off the "make material" idea. לְהַגְשִׁים is causative: you make the dream real. לְהִתְגַּשֵּׁם is its reflexive mirror: the dream makes itself real, it comes true. There is no everyday pa'al "to rain" from this root (Hebrew uses other phrasing); the concrete גֶּשֶׁם stays a noun.
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.