ご面倒をお掛けして申し訳ございません。 means "We are sorry for the inconvenience." but what is the meaning of 掛ける in this context?
The meaning of 掛ける in that context can be thought of as "to burden".
Of course the English translation then looks like it should be "Sorry to burden you with trouble/nuisance", which isn't natural in English. It works in Japanese though, and strengthens the feeling of inconvenience.
Note that 掛ける is one of those words that doesn't really have a single representative English equivalent. In fact, EDICT lists no less than 23 potential English translations.
If I had to give a more general description of the word used in this kind of context, I'd say you might like to think of it as describing an action/effect being inflicted or imposed on someone/something. But please use this with caution, as it's not always right.