dark, disappear
晦 centers on darkness and obscurity, both literal and figurative: something dark, dim, or hidden from view. The idea of disappearing follows naturally from being swallowed by darkness.
晦 combines 日 (sun) with 每, which likely contributes the sound and carries a sense of darkness or obscurity. The character originally referred to the last day of the lunar month when the moon is dark.
The sun 日 is beside 每, which looks like a person with a head covering. Imagine the sun being covered, making everything dark and causing things to disappear into shadow.
For カイ, picture a dark sky where the sun disappears behind a thick layer of clouds, like a dark 'kai' (invented sound for a heavy, dark blanket). The kai sound cues カイ while the scene matches the kanji's meaning of darkness.
New Year's Eve
to conceal (oneself); to hide; to abscond
the last day of the year; New Year's Eve
the world is covered in darkness; all is plunged into darkness
darkness
last and first days of the month
ambiguous; obscure; equivocal
concealing one's talents, position, status, intentions, etc.
concealing one's talents, status, intentions, etc.; maintaining a strict incognito; self-effacement
propensity to efface oneself; being prone to conceal one's talent