This may or may not also have been spurred by Nintendo losing marketshare with the Nintendo 64, and wanting to make sure their handheld line didn't stagnate.
Either way, they learned the lesson of the N64, and this system was easy to develop for. Nintendo basically enhanced the Game Boy parts, and made a souped-up NES. This made sense, since the Game Boy was similar to the NES already.
It also follows the design paradigm. Its about the size and weight of the Game Boy Pocket, and is just as durable. Its launch price was in between the launch price of the Game Boy and the Pocket. Its battery life is about the same as the Pocket's, and best of all, it was backward compatible with Game Boy games, with a handful of 10-color palettes built in to give those old games a hint of color. However its inability to use the color palettes in Super Game Boy-enhanced titles drew small amounts of criticism.
The system was a hit, selling so many systems that Nintendo delayed the release of the Game Boy Advance a few months (since the Color's main processor was meant to be included to ensure backwards compatibility). The totals sales is unclear, since Nintendo folds the total with the original Game Boy for some reason, but the fact that it was delayed meant it had to be selling a lot.
An interesting addition was the inclusion of an IR port, which could allow for wireless data transfer between two systems. It wasn't as large-scale as using the link cable though, and could only be used for very minor things. As usual for the Pokémon series, Pokémon Gold and Silver were probably the most well known examples of this features, allowing two players to "Mystery Gift" with each other once a day, basically creating random items for both players. The games could also communicate with a virtual pet, which could get the player a variety of items.
Unfortunately, the system lasted only a few years before retiring, due to Nintendo moving on to the Advance. This makes it probably the most successful system ever to have such a short cycle.
* CPU is an enhanced version of the Game Boy's Z80-compatible running at 8 MHz (almost twice the speed of the original Game Boy's CPU, as well as the stuff that was removed from the the Game Boy's CPU was put back in for the Game Boy Color's CPU).
The structure allows the system to play original Game Boy games.
* Sprites and resolution are the same as the Game Boy.
Backgrounds can be more detailed in GBC-only games due to additional memory for tiles.
Up to 56 colors on screen (eight 4-color background palettes and eight 3-color sprite palettes), out of 32,768 total. Note that this is not really a hardware limitation, but rather palette memory limitation. If the palette is changed every scanline, more colors can be achieved.
* 2 AA
Battery life just over 20 hours.
In short, the system can't exactly run rings around the NES, but it can handle graphics just as intense without slowdown or flicker.