Well a picture is worth a thousand words so here's 3,000
This shows the relative difference of saturation between sRGB and AdobeRGB. Both sides have the same RGB values, the gradient runs from 255 to 127 but the different colour space means the result is very different. The AdobeRGB version is "clamped" at the top because the saturation is beyond what my monitor can display.
This shows how the image would look when printed by DSCL compared to the sRGB version. Again the difference in saturation is clear, the colour space of the printer is smaller even than sRGB. Hopefully this makes clear why AdobeRGB doesn't give you "more colours", it's how big the colour space of the output device that's important. If that can't handle the colours you'll never see them and the values set aside for those colours are wasted. By using a smaller colour space that's closer to the output device you're making better use of the values for smoother gradients.
Another common problem with AdobeRGB is when they're posted online, or even printed at a lab such as Photobox or DSCL. Unless the image is converted to sRGB first you end up with this:
The image was originally in AdobeRGB and only the middle part has been converted to sRGB, the outsides are AdobeRGB but being displayed in sRGB without conversion (note in particular the pink stripe on the little girls dress). This is what would happen if the image was printed by a lab such as PhotoBox, DSCL, Boots, etc. as they expect the image to be in sRGB and don't colour manage. The same happens if you view the image in a web browser or other application that isn't colour managed because the s/w isn't converting the RGB values so they are correctly displayed on your monitor.
Hopefully this makes things a little clearer. Colour management, profiles and calibration is a complex subject but get your monitor calibrated stick with sRGB throughout your workflow and you can't go wrong - images will look good on screen (and pretty close even in an unmanaged app) or print. When working with raw files the colour space for the image is decided by the raw app, not the camera setting so make sure it's exporting images in sRGB as sometimes this setting can be hidden - Adobe Camera Raw for example defaults to AdobeRGB but the option is tucked away.