Some people say it dates back to the Cavaliers who wore thigh length boots. When drinking, rather than stepping outside to relieve himself, a Cavalier apparently had the option of doing so into his boots. Thus, "filling his boots" meant he could drink all he wanted without leaving the table.
Apparantely, at the HMS Victory museum in Portsmouth UK, you can buy a thick leather cup lined with pitch. This is a replica of the sailor's mug used on board in Nelson's time, and it was used (among other things) for the rum ration when issued. This cup is called a "boot", and when things were good and you got an extra rum ration, sailors were told "Fill Yer Boots"!