What does romance have to do with horses?

Well, the word romance has a number of meanings:

– A story relating to chivalry; a story involving knights, heroes, adventures, quests, etc.
– An intimate relationship between two people; a love affair.
– A strong obsession or attachment for something or someone.
– Idealized love which is pure or beautiful.
– A mysterious, exciting, or fascinating quality.
– A story or novel dealing with idealized love.
– An embellished account of something; an idealized lie.
– An adventure, or series of extraordinary events, resembling those narrated in romances.
– A dreamy, imaginative habit of mind; a disposition to ignore what is real.
– (music) A romanza, or sentimental ballad.

The first meaning provides the connection to chivalry, a word which comes from Middle English chivalrie (quality of being knight), from the Old French chevalerie (knighthood, chivalry, nobility, cavalry), from chevaler (knight, horseman), from the Medieval Latin caballarius ‎(horseman, knight), from the Vulgar Latin caballus ‎(horse), which meant ‘work horse, pack horse or nag’ in Classical Latin, possibly from the Gaulish caballos (horse), or from an Old East Iranian dialect.

The word romance comes from the Middle English romauns, roumance, from Anglo-Norman and Old French romanz, romans (the vernacular language of France, as opposed to Latin), from the Medieval Latin rōmānicē, from the Vulgar Latin rōmānicē ‎(in the Roman language), from the Latin rōmānicus ‎(roman) from rōmānus ‎(a Roman).

Sources: Wiktionary, Online Etymology Dictionary, Degruyter.com