Anamorphic is a term that describes a particular process that, in this context, only applies to DVDs. Take a hypothetical 16:9 film. This film is compressed (squished horizontally) to a 4:3 ratio, and stored in this format on the DVD. The DVD player then stretches it back out to 16:9, restoring the proper ratio and allowing it to fill the whole screen of a widescreen TV. This explanation is a vast simplification, but that's the effect.
A non-anamorphic 16:9 film is not compressed or stretched, but still must be stored in the same way. This means that extra black bars are needed at the top and bottom, and the DVD player does not stretch the image and does not fill the widescreen TV. You are left with a 16:9 image surrounded by black bars on all 4 sides. The aspect ratio is correct, but there is more of your TV that you could correctly fill without breaking the aspect ratio if the movie had been anamorphic-formatted.
With Blu-ray, the native storing format is 16:9. So no compressing / stretching is needed. The Blu-ray player simply outputs the stored 16:9 image, and it fits properly on your widescreen TV.
If anamorphic HAD been used for Blu-ray, it would be to store 2.39:1 films in a compressed 16:9 form, for later stretching on potential native 2.39:1 TV screens. Such screens don't currently exist, but if they did you'd face the same problem as with non-anamorphic DVDs on a widescreen TV, except now it'd be non-anamorphic 2.39:1 films on a "super" widescreen TV.