They have possibilities anywhere you join two pieces of leader/tippet together. Rings allow you to avoid nail knots, surgeons knots, etc. and eliminates the need to cut back you leader every time you re-tie tippet. They also provide a convenient and easy point of attachment for droppers in multi-fly rigs.
Personally, I really like using them on the end of a sighter in a Czech rig so that I preserve the full length of the sighter when re-rigging. I've also used them on the end of a fairly heavy butt section for indicator rigs, however they are a little small for use with heavy mono. Not sure using them with anything above 20lb test is feasible. The hole is pretty small.
Notice that the video you linked also shows small swivels. Probably a better choice when using heavier mono. Either way, rings or swivels, they can make rigging easier and re-rigging faster. You can also easily do without them. Some people might have a problem adding "hardware" to their leaders but they already fish an indicator, split shot, and a weighted fly. It's really up to you, these aren't a "must have" for most people.