How does 驕's 2021 meanings appertain to its semantic component 馬?
A horse might not directly represent 'arrogance', but status, wealth, power, elite, proudness, and unruliness are all potential cause of arrogance

Owning horses was a sign of wealth and power in the time most of the population were farmers

The number of horses you were allowed to use for a carriage indicated your status, No matter how wealthy you were, if you were not a noble, you could not have a four horses carriage and only the emperor could use a six horses carriage

Only Elite warriors ride horses

Wild horses are proud, unruly


Notice: 'proud' and 'arrogant' are very similar, differ mostly in degree of outward expression. 'proud' is more likely the original meaning of 骄 and 'arrogant' is a logical extension