I think in one of the videos they showed them being dug up from either razed cities or old city ruin sites, but I haven't seen a concrete explanation. How exactly do I acquire artifacts to fill my museums et al?
Artifacts are gained by using Archaeologists. The Archaeologist unit is unlocked by the Archaeology technology in the Industrial Era, and also can only be built in cities that have built a University.
Once you have the Archaeology technology, even before you make an Archaeologist, you will be able to see basic antiquity sites. Move an archaeologist to one, and they will be able to work the tile as workers do other tiles. This takes a handful of turns (depending on game speed), and when they are done you will have the option to gain an Artifact, or to create a Landmark improvement on the tile.
If you see an antiquity site in another civilization's territory, you can still work it, but doing so will potentially anger the other civilization, in much the same way as converting their cities does when they have their own religion.
Lastly, if there are several other civilizations working archaeology sites, you can try getting the Exploration Social Policy Tree. Having all policies in the tree unlocks hidden antiquity sites, which function exactly the same as regular antiquity sites, but cannot be seen nor worked by civilizations without the full Exploration tree, which means there may be less competition.
Once you discover Archaeology, a new type of Bonus Resource will appear on the map -- Antiquity Sites!
If you build an Archaeologist (also unlocked with Archaeology) and move them to an antiquity site, they can perform an archaeological dig that will give you your choice of an Artifact or to create a landmark on the site.