Please explain what is meant by the cuticle being left on!

Well, here's my understanding of it. (Anyone jump in here if I'm off-base.)

The cuticle (the protective outer layer of the hair) is sort of like a series of overlapping shingles (like on a roof). Because it's the outer layer, it's the first layer to get damaged and break down from bleaching, relaxing, hot tools, or mechanical friction. Once the cuticle is damaged, it can never be repaired, really. You can use conditioners and such to make the hair look and feel better, but ultimately, the damage will remain.

Along these lines, extensions hair is usually quite processed to achieve the colors and textures that would be desirable to the client, so in those cases the cuticle is generally quite damaged or basically gone. If that's the case, it may be advantageous to remove it altogether.

The other (and very common case) where you'd want to strip the cuticle is when the hair is collected from brushes (and most extensions hair, imo, is). Because all of the hairs are facing every which way (because they've been collected at random), the cuticles (or what's left of them) will tangle like crazy. In that case, too, it's advantageous to remove them. This is usually done in an acid bath.

Ideally, all hair would be full cuticle and facing all the same direction, but since this is very rare, cuticle stripping makes more sense than having the unruliness of mangled cuticles catching on each other all the time. Of course, any non-cuticle hair (because it lacks that outer layer) will not last very long. It will be prone to breakage and split ends and will lose its sheen (usually a silicone coating) rather quickly.

I hope that helps. :)