I have a general idea of what digipaint is, but I couldn't find a lot of the sources that gave me this idea since it was mostly stuff I just picked up in passing. I did find one blog that went over the process and more or less confirmed what I had thought; scroll way down to 'Compositing / "filming"' to see where they discuss it.
This is the gist. Up until about 1999–2000, anime was created by drawing line art on a clear plastic cel, then hand-painting that cel, with a brush. Characters and backgrounds were drawn on separate cels; character cels were placed on top of background cels and the animators would take pictures of each frame, stringing them together into a film.
Sometime around 2000, animators started coloring the line art using a computer. This is basically similar to making art in Photoshop, though according to the blog they use a program called RETAS! Pro. Backgrounds can easily be drawn separately and added later on a computer; if you've ever used Photoshop, you know how easy it would be to draw a character and a background as separate images and combine them later as layers of a single image. The filming process also started to be done on a computer; the animators create the frames digitally based on the line art, then use the computer to string the frames together into a "film".
Digipaint refers to this process of coloring a cel on a computer with a program instead of by hand with a brush and paint. The blog and this other one discuss some of the effects the new process had on the anime industry: since it was faster and easier to color frames and composite character and background art on a computer, shows could have more frames, which led to smoother animation that wasn't reused nearly as often. It was easier to add fine details, so character art became more detailed. (I assume this was because of zooming and the precision of a graphics tablet, plus the ability to undo mistakes.) It was also easy to add in 3D computer graphics, so things like special effects and machinery, which are much easier to create with CG than hand drawing, started to be done in 3D.
On the other hand, it took a while for animators and art directors to get used to the different character of digipaint compared to traditional paint, which led to the ugly color palettes, blurriness, and flatness that you see in a lot of early digipaint shows (from 1999–2004 or so). These problems are especially noticeable in some of the lower budget hentai from this era, where characters and background art look like they exist on two separate 2D planes rather than in a single 3D space, making the animation resemble a paper cutout puppet show.
As far as I can tell, digipaint is still commonly used today; animators have just gotten a lot better at using it. Last Exile is another show from the early era that I remembered looking a lot better than it did. Like Haibane Renmei, it was a show with amazing, imaginative visuals that ended up looking like hell because of a hideous color palette, rampant blurriness, and primitive CG integrated poorly with the 2D animation. I haven't watched Fam: The Silver Wing, but it would surprise me if it didn't look much better than the original.
Some shows have moved towards cel-shaded 3D animation. This is a technique where everything in the show is created in 3D CG, but through the use of filters and such, it almost kinda sorta ends up looking like traditional 2D animation. Cel-shaded 3D has been around for a while in video games, but it's just now in its early days for anime and suffering from problems just like early digipaint. However, it will probably end up replacing digipaint, since it's cheaper (just render a 3D model from different angles. It's almost like having actors again!) and getting better all the time. (Cf. Arpeggio of Blue Steel.)