N.B. The screenshots and descriptions are for the iPhone/iPad version.

This is a history per horse of the activities undertaken via the 'Lists' option on the Home Screen. (Apart from riding, activities can include lunging, hand walking, horse walker etc.)
We call this "Exercise" rather than "Training" on purpose — it covers all physical activity, not just schooling. For example, a horse recovering from a leg injury might be walked in hand by a groom as part of its rehabilitation: that's exercise, not training, and it's recorded here too.
If you start a ride list for a particular day, allocate a horse to a rider or handler and enter any notes, the exercise history for that horse is automatically updated here.
Although these entries are usually populated automatically from your Lists, this is not a read-only history — you can also add or edit an entry directly using the Exercise link on a horse's home page. That is useful for recording an ad-hoc or unplanned session (for example a spontaneous hack, or a rehab walk) that was not part of a List. If you use Lists as your focal point for horse exercise, there is no need to enter it separately.