Yup, those are their exercise yards. From Mickey & Friends you could also see their stables and grooming areas. Disneyland had a huge setup for their horses, but they only kept less than a dozen.
A family member of mine owns horses and is involved in national Dressage competitions, and when she saw the stables area from Mickey & Friends she commented to us that they had a very fine looking facility large enough for dozens of horses. But Disneyland only had a handful of horses, and you could only ever see one or two horses in the facility at any one time.
The horse-drawn streetcars continue daily operation today, until PETA aims their protest movement at Disneyland and WDW, which is something I'm waiting for. Where do Disneyland's horses live now? I forget where they moved them to.