These animations look like traditional pixel art to me. They can be created by artists in any raster-oriented image editor by placing each pixel individually. This art style does not require complex software. There is specialized software available like Asesprite, but the basic techniques can just as well be learned and applied in any general purpose image editor, even in a very simple one like MS Paint.
There are some filters available which can turn 3d renderings into convincing looking pixel art. One of my favorite applications of this is from the game Saints Row IV where they suddenly turn a high-budget 3d game into a retro pixel art beat-em-up. You can tell it's not hand-pixeled, because it applies the extensive character customization options to the player-character. Doing every single outfit again in pixel art would have been far too much work just for this short sequence.
But I don't believe that such a filter was applied in the example animations from this question. The art looks far too traditional to me when it comes to "pixel economics" (using exactly as many pixels as needed for every detail). It also does some very neat cloth and hair animations which would be very tedious to do that way in 3d but are actually not that hard (still not trivial, though) to do for an experienced pixel artists.
Pixel art is an artistic medium which is easy to learn, but hard to master. In order to be able to create animations in that level of quality usually requires several years of study and practice. Although a lot of basic principles in art are of course transferable between different techniques. Color theory, the 12 basic principles of animation, human proportions and body language or perspective are universally applicable. So if you already studied those subjects in the context of other visual mediums, then your journey to becoming a decent pixel artist can be a lot shorter.
Why is dithering used in pixel art?
One reason is nostalgia. Old gaming systems only had a very limited color palette available. But dithering allowed them to create blends between two colors, artificially expanding the available color palette. This actually worked a bit better than today because the carthode-ray tube monitors and TVs of that time produced less sharp pixels than today's LCD displays. Today this would be unnecessary, because pretty much any gaming device supports 32bit colors. But nevertheless dithering is still used as an aesthetic choice to emulate the classic look of 90s video games.
Another reason is that dithering is a good way to simulate texture. Here is an example from a well-known pixel art tutorial which compares cell shading to dithering:
Note how random dithered shading makes the objects appear a lot more rough and interesting.
The examples in the question actually show a more regular, checkboard-style dithering (very visible on the skirt of the character in the first animation). I personally don't like that style very much, because it looks artificial and unnatural to me. I much more prefer placing pixels more randomly (like in the harpy on the 3rd image) or if I want the material to look smooth, use bands of color like on the skirt in the 2nd image. But that might just be a personal preference.
