The adjective exceeding comes from the verb to exceed, which means "go beyond." The massage was given with exceeding care — it was worth every cent.
A parent feels exceeding pride at the graduation of a kid. You might also say that they feel "extra," if that helps to clue you into the meaning of exceeding, but it does have a slightly different meaning. If you were to have "extra" pride, then you'd have more than you needed. You might give the "extra" to someone else. To feel exceeding pride, though, means to have more than you might have expected. It's another way of saying something is exceptional in whatever way you mean.