extra horse, imperial son-in-law
駙 centers on a supplementary or attached horse, and by extension an imperial son-in-law, who was historically seen as an attached member of the imperial family.
駙 combines 馬 (horse) with 付 (attach), suggesting an attached or extra horse. The imperial son-in-law meaning likely developed as a metaphorical extension of this attached status.
A horse 馬 with something attached 付: picture an extra horse tied to the main one, or a son-in-law attached to the imperial family.
For フ, imagine the extra horse making a soft "fu" sound as it follows along: fu -> フ, like a gentle breath of an attached horse.