Healing skills scale with magicka

This has been explained in one issue of their Ask us Anything series.

Do attributes (health, magicka, and stamina) affect more than just HP and your magicka or stamina pool (like resistances, speed, etc.)?

Yes. Your attributes will affect certain other stats. Some of these will be exposed, but others will be more behind the scenes. As an example, magicka affects not just how many spells you can cast, but their overall efficacy as well.

Quick rule of thumb:

• If your skill consumes magicka, then magicka will increase its number values.
• If your skill consumes stamina, then stamina will increase its number values.

Also note that - as this kb article explains - most weapon abilities will consume stamina, staffs being the exception as they consume magicka.

Do bows require stamina to shoot? And do Restoration or Destruction staves cost Magicka to use? Or is there some other resource used for staves?

Normal attacks with weapons don't require stamina or magicka. If players learn abilities for that weapon, these abilities will have a cost associated with them. Almost all weapon abilities, including those for the bow, will consume stamina. Staves are the exception—their abilities consume magicka.