"I know how to ride a horse."
Translation:Je sais monter à cheval.
'Monter un cheval' is the single action of mounting a horse, and whilst a grammatically correct construction, does not indicate the ongoing ability to riding a horse, which is 'monter a cheval.' Same as in Spanish 'montar a caballo,' or Italian 'montare a cavallo.' Perhaps DL could better phrase the English sentence as 'I know how to ride horseback.'
It is never wise to expect prepositions to line up in meaning from one language to another. They are rarely very logical. You can run up a new dress, run down a pedestrian, run over your notes, run out your fishing line, run through a scenario. None of that is particularly logical.
This discussion has stirred a memory that the Teach Yourself '50 Ways to Improve Your French' advises care in using 'monter' because of its sexual connotation. It gives as examples: 'Tu montes À cheval' and 'Le cheval monte la jument dans le champ'.
Just for interest, that very helpful little book also says that 'monter' (without à) can also be used for assembling things (like IKEA furniture) and for taking things upstairs.