Flake could have given you reams of technical and equally useful info regarding your question but I note she hasn't been around for a number of days so in her absence you'll have to put up with me.
They say that the letter 'L' stands for luxury although whether this is true or not I don't know. The fact remains however, that these lenses are
supposed to be produced to a higher quality level including better weather sealing. That said, I have italicised the word supposed because sometimes manufacturers e.g. Tamron and Sigma produce an equally good lens and at a cheaper price. In this respect it is always worth while doing a good search when considering a particular focal lens length lens to make a comparison.
There is absolutely no reason why an L series lens shouldn't be used on a crop sensor camera - in fact, sometimes with better results than on a full-frame camera. All lenses deteriorate in quality (some more than others) at the periphery of the lens. A full-frame camera with it's larger sensor and larger field of view (FOV) utilises more of the lens and so those who own full-frame cameras more often than not advice that only the best quality glass should be used. On the other hand, the smaller crop factor sensors with their narrower FOV don't take advantage of the full area of the lens and as a result the last point isn't quite so critical. This applies to all lenses of course, not just the L series.
Don't forget that there is another series of Canon lens - the EF-S range which are specifically produced for crop sensor cameras and which cannot be used on full-frame sensor cameras.