I would translate the grammatical word casus (whence English case) as "a fall". And, indeed, the German word is Fall, Dutch naamval ("name fall"). Why is this word used for the grammatical function of nouns?
The Romans already used casus thus. But its connotations were "occurrence" and "accident" (which happens to be cognate with case), which do not seem related in meaning at all to the grammatical sense.
There must be some kind of metaphor at work. During my studies, we were told to imagine the casus rectus (undeclined or nominative form) as a stick or line standing up straight, the obliquus as slanted at an angle. Indeed, the verb declino was also used by the Romans to mean "1. slant down; 2. decline grammatically". English inflect also fits.
I can understand this metaphor, but it seems somewhat odd. Why would a grammatical function be standing up?
In addition, it seems to conflict with casus as a metaphorical fall. A fall is downwards, and how does a slanted fall work—let alone a leaning ("declined") fall?
I believe there is a conexion between Greek κλίνω and declino, as the former is also used for grammatical declension. The same applies to ὀρθός "rectus" and πτῶσις "case", which apple the same mataphors. Is the Latin just a loan translation? In that case, how did Greek come up with falling and leaning at the same time?