What's the logical/etymological connection between 'to decline' and grammatical 'declensions' of nouns?

The verb decline comes from the Latin declinare 'to turn away, bend downward, change the form of a word'. The different forms of a noun are called case. Case is from the Latin casus 'fallen' which is a translation of the Greek πτωσις (ptosis) 'falling, fall'. The idea is that the nominative, or naming, form of the noun is the normal one and the oblique cases (the genitive and others) are falling away from that normal case. This sort of thing used to be called accidence. Hope that helps.

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the Greek πτωσις (ptosis) 'falling, fall'

In ophthalmology the word for a drooping or limp, half-closed upper eyelid is ptosis.