How do spirits heal? I assume they can be magically healed.

And I guess first aid and medicine are useless.

But, do they heal any other ways? I seem to remember that, in previous editions, the summoner could use services to have the spirit heal. Is there any mechanic like that now?

well, in previous editions they healed when they returned to their metaplane. so you could send them to heal as a service, iirc.

if you're not talking about a bound spoirit, just release it and summon another in it's place.
With bound spirits it's a little trickier, you could have a rotating stock of them sending them back to their home planes to be healed and calling fresh ones to replace the injured. This would become quite expensive quite quickly given the cost of binding those little suckers. I don't know if the current RAW allows them to come back healed immediately or if a certain amount of time must pass.
If I were gming I might allow someone who had the specialization in natural healing use medicine or first aid; but only if they had the specialization