Origins
Latin habēre, 'to hold', is one of the most versatile verb roots in English. Attach a prefix to it and the meaning shifts with the direction. Exhibit — ex- ('out') + habēre — means to hold something outward, to present it for inspection. Inhibit — in- ('in') + habēre — means to hold something inward, to restrain. Prohibit — pro- ('before') + habēre — means to hold something before it can advance, to block.
The word entered English in the 15th century, initially in legal contexts. An exhibit in court was a document or object held out for the judge and jury to examine. This remains standard legal terminology: 'Exhibit A' is the first item of evidence presented.
Semantic Evolution
The cultural sense — a public display of art or artefacts — developed in the 17th century. The Great Exhibition of 1851 in London's Crystal Palace cemented exhibition as the word for a large-scale public showing.
The Italian cognate esibire carries an additional meaning absent in English: to perform. An Italian musician esibisce on stage — holds out their art for the audience. This performance sense reveals the original gesture embedded in the word: someone extending their hands, offering something for others to see.