They say that AdobeRGB has more colors than sRGB color space.
What do they mean? Should they both have 16,777,216 colors (because they're both RGB color spaces)??
Adobe RGB also tends to be more red-ish, it's often used when printing stuff, sRGB is more common in the digital world with monitors and digital-media. They are different color-spaces, how many values they can represent it's something that is implementation-dependant, it's not the main difference between them.
In the color-world there are color-models and color-spaces, this 2 are both RGBs so they belong to the RGB color-model family but they are 2 different color-spaces.
The difference here is about how the arrange color values in the space and based on what, to really understand this you should study how a printer works from a color standpoint and how a display works.
It's like how the Mp3 compression works, you have an uncompressed audio source, you want to compress it or in other terms, to represent only the most significant Hz, so you need to make a choice AND you need to give an algorithm, a rule; in the Mp3 case the rule is simply what Hz the human ear can catch easier than others, you discard all the ininfluent or unheard Hz and you only keep the most important ones. In the RGB case you don't have audio but you got colors, you don't have Hz/waves but color-values/hues, and you always got a finite space, meaning that you need to make a choice because you can't store infinite values, so you basically make a choice based on what are the studies behind this 2 color spaces and in a nutshell the AdobeRGB is best for printing and sRGB is for TV, monitors and everything similar. To undestand how they works just read the papers, it's impossible to explain this in a short answer but there are many resources around the web.
Check out this article http://www.kenrockwell.com/tech/adobe-rgb.htm which basically says that they both occupy the same range, but due to software the Adobe RGB can be unpacked and then further expanded.
Further reading about the 2:
So, it would appear that the colours available to RGB is much greater than what is offered by either sRGB or Adobe RGB and that Adobe applies a compression technique as well as choosing to select a different range of colours (compared to sRGB).