I haven't looked into GRRLIB, so I can't comment on the documentation, but in general the importance of good documentation is overlooked in the software industry...
GRRLIB must use GX under the hood (GX is the only graphics library), so based on what Daid said I'm guessing somewhere we have the following call:
GX_SetTevOp (GX_TEVSTAGE0, GX_MODULATE);
This sets the texture environment (TEV) operation; there can be several stages in a TEV, and the operation determines how the color is combined at each stage. GX_MODULATE means multiply the color and alpha channels, so as Daid said the input color component values (from the color parameter) are normalized and multiplied by the existing color component values (from the texture).
So if your texture has an alpha of 0xFF and you multiply it by a color that has an alpha of 0x77 = 119 ==> 119/255 = 0.47, the pixels output by the TEV will have an alpha of 0.47. The blending you describe happens after the TEV completes. When the output pixel is drawn into the EFB (embedded frame buffer), it's blended with the pixel that's already present. The standard blending algorithm is:
(1 - alpha) * current_color + alpha * new_color
So the blended pixel will be 47% of the new color (the texture color) plus 53% of the existing color (the background color).
Understanding GX will certainly help you understand all these libraries that are built on top of it. Unfortunately, GX documentation is about as bad as it gets, but don't panic.
1. Reading patents is legal. If you can find legal documentation describing how Nintendo's official GX library works, then it will help you understand the homebrew GX library.
2. GX is pretty similar to OpenGL. If you know how something works in OpenGL, then it's easier to understand in GX.
Finally, if you notice that documentation is lacking, I would encourage you to get in there and donate some documentation; that way the whole homebrew community can benefit.