What is "layering flavors"? What does it accomplish and how do I do it?

It is mostly a fancy way of saying that they are combining flavors. There are no solid, physical layers involved anywhere.

Still, there is a reason why the "layering" metaphor is more apt than simply saying "combining". Flavor is mostly about aroma, which leads to two aspects of "layering".

First, aroma is not perceived all at once when you bite off. You first notice the most volatile smell notes when the food enters your mouth, go through a kind of "middle" and only at the end, while swallowing, you notice the "heavier" flavor. So, when seasoning, you can work with food such that you don't mix up too many flavors in one of the three aroma "layers", but also to make sure that there is something noticeable in each of the three.

Second, people are accustomed to some flavors being present as a metaphorical "background" taste. It can happen that, if the expected aroma is absent, you can add all the spices you want, and the eater will still experience it as underseasoned. This is what is happening in the example you cited: the "base layer" are the onion and garlic, which are probably the standard for the dish, and then you can take that dish in different directions by your choice of additional herbs and spices. In that case, you can see the metaphor as akin to clothing - once you have a basic shirt on, you can always make the outfit nicer by layering a scarf, jacket or jewelry on top of it.

All in all, it is not "all about layering flavors". It is just an expression which helps some cooks go about creating harmonious flavor combinations. If it works for you, use it. If not, learn to think about your seasoning in other terms.

If you could link the video with the quote in question that would be helpful.

In the case of onion and garlic from your example, I would say "layering flavor" is just a way of saying "bring out the best flavor(s) from each ingredient by appropriately adding them in a certain order". Onions have quite a bit of moisture in them that needs to be reduced before caramelization can occur. Garlic on the other hand is quite dry and turns bitter when heated for too long on too high heat.

So, by first allowing the onions to cook on their own and developing a nice flavor and then adding the garlic for a comparatively short amount of time, you get both the flavor of the caramelized onions and the garlic. If you added both ingredients at the same time, you would need to deglaze your onions earlier in order to prevent the garlic from burning, thus missing out on the caramelized onion flavor.

Of course, the method of this "layering" differs from dish to dish, depending on what you want to achieve. A lot of stir frys try to not overcook vegetables so they are still a bit crunchy, while a stew that cooks for a long time goes for the opposite.

It's also about adding flavors at each stage of cooking. Each ingredient added should be seasoned, rather than dumping everything into the pan at once and adding seasoning. Example: Saute onions first, season with salt and pepper. Add diced carrots, celery, and peppers. Season. Before sauteing the protein, season well with salt and pepper and desired herbs. All layers of the dish are seasoned, resulting in a more flavorful dish. Each bite is seasoned, not just the stuff at the top of the pan that caught the salt and pepper.

As @rumtscho has explained in his accepted answer, it's more a style of language than anything else.

In your particular example, the reason why onions are added first, and then the garlic, is that onions can and should be heated much longer than garlic - garlic burns or turns bitter quickly; throwing them in at the same time would either undercook the onions, or overcook the garlic.

Regarding the question about how to do it - basically, for me the trick is not only to focus on the main ingredients, but also to always think about what else could "work" in there. For example, I usually put a few drops of lemon juice and a very tiny pinch of sugar into most sauces - you don't taste either, the sauce won't have a citric or sweet taste in any shape or form, but it somehow tastes more interesting.

That would be a typical thing someone might call "layering" and it just means that aside from the main ingredients there are very minor bits of other tastes in there.

One perspective I've encountered looks at specific types of ingredients when discussing 'layering.' For example, 'layering salt' is about adding saltiness to a dish from different sources: e.g., seasoning first with salt and finishing the dish with Parmesan.

'Layering fat' could be adding butter to the first stage of cooking, then olive oil near the end.

I find this use quite intuitive (and distinct from 'combining') because the order matters—i.e., cooking in oil and then adding butter is different from the converse.

The key is that different sources of an ingredient type can be added at different stages, each adding nuances to the flavour and texture.