I reached out to Premier Support to get this question answered, as I couldn't find any documentation about this feature either.
On Demand Refresh option can be activated only once per sandbox per refresh cycle, and is useful if a business needs to refresh a particular sandbox more than once in between its refresh cycle.
On Demand Refresh can be activated against a sandbox by raising a case with Salesforce Support, who would then escalate to Tier2 and Tier3 to get their own internal approval. Once internally approved by Salesforce, the On Demand Refresh check box would be ticked, to indicate that you can refresh a particular sandbox again in between its refresh cycle.
If you have a full copy sandbox then you can only refresh this sandbox once every 29 days. The sysAdmin refreshes full copy sandbox on May 1st, and then the business realizes 3 days later that they need to refresh this sandbox again.
Now, you won't be able to refresh this full copy sandbox until May 30th (provided that full copy sandbox actually did complete its refresh on May 1st). This is obviously problematic for the business. So a solution is to reach out to Salesforce Support and request the On Demand Refresh option, which would enable the business to refresh only one more time from May 4th to May 30th (the refresh cycle).
When May 30th is reached, then you would be able to refresh the full copy sandbox normally. But again, if you wanted to refresh full copy sandbox one more time in between the new refresh cycle (May 30th to late June), you would again need to reach out to Salesforce Support to get the On Demand Refresh option activated.
I assume that On Demand Refresh can only be activated via Salesforce Support because refreshing sandboxes is a relatively computationally intensive process (especially for full copy or partial copy sandboxes). I assume for the same reason that On Demand Refresh can only be granted once per refresh cycle, so the most you could refresh a full copy sandbox is twice per refresh cycle (once normally and once via On Demand Refresh).