fine carries two families of meaning that diverged from a single Latin root: one describing something of superior quality, delicacy, or thinness, and another naming a sum of money exacted as a penalty.
The adjective arrives in the mid-thirteenth century from Old French fin, "perfected, of highest quality," which took the sense from Latin finis — "boundary, limit, end," and figuratively the peak or utmost of a thing. From that core sense the word fans out along two axes. One is excellence: fine wine, fine art, a fine performance. The other is smallness or subtlety of scale: a fine point, fine print, fine sand, a fine distinction, and by extension the sharpness of an edge, attested from about 1400. A third and much weaker sense now dominates conversation — fine as merely satisfactory, requiring no further attention. That weakening is what makes "I'm fine" so unreliable a report; the word has drifted far enough from excellent to serve as a way of closing a subject rather than describing a state.
The noun took the other branch of finis, the literal end. Around 1200 it meant a termination or settlement, and in Medieval Latin finis had already come to mean a payment made in settlement. By the mid-fourteenth century Anglo-French used it for a sum paid to be exempted from punishment, and the modern sense — a monetary penalty imposed for an offense — is recorded from the 1520s. The verb followed the noun, meaning "to pay as a ransom or penalty" in the late thirteenth century before inverting by the 1550s to the sense it holds today: to punish by imposing a payment.
As a backronym, F.I.N.E. is glossed "fucked up, insecure, neurotic, and emotional" — the reading Aerosmith gave the title of a track on their 1989 album Pump, and one that circulates widely in recovery culture as a name for the state a reflexive "I'm fine" conceals. Variants substitute needy for neurotic.