Why is my count returning a syntax error?

You need to name your subquery; try putting something after the final parenthesis:

ALTER PROCEDURE [db_owner].[get_checkin_likes]
@likes integer
AS
Select count(*) 
from
(
  select checkin_id as Checkin, users_id as 'Liked by'
  from checkin_likes
  where @likes = checkin_id
) a

I probably should mention that when you run this query you will only get the count back. I'm not sure of your use-case here, but you can do an OVER and return everything with a count at the end

EDIT:

I would personally do this and have it all in the one record set:

ALTER PROCEDURE [db_owner].[get_checkin_likes]
@likes integer
AS

  select checkin_id as Checkin
    , users_id as 'Liked by'
    , COUNT(checkin_id) OVER () AS total_rows, 
  from checkin_likes
  where @likes = checkin_id