The core idea is coming near. The first letter נ is weak: it drops out of the pa'al infinitive, which is why "to approach" looks like לָגֶשֶׁת, with no נ in sight. The same root, in the hif'il, means to bring something near to someone, hence to serve food or to hand in a form. From there the family gives you access and a tray (what you bring food on), the quality of being approachable, and, when two things approach too hard, a collision. The disappearing נ is the puzzle of this whole family. 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 weak initial נ makes the pa'al look irregular: it assimilates and vanishes, leaving לָגֶשֶׁת. In the hif'il it also drops, giving לְהַגִּישׁ with a dagesh in the ג that "remembers" the lost נ. The hitpa'el here is not reflexive of approaching but of clashing, the collision noun below. This is normal behaviour for first-נ roots.
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.