It is implied (heavily) that Hagrid's umbrella contains the pieces of his wand which was broken after his expulsion from Hogwarts. However, there are many examples of wands being broken and they either don't work at all or work very poorly.
Even though Hagrid isn't an excellent magician he is still able to perform simple spells using his umbrella. How is it that his umbrella works so well for him?
Answer
All of the broken wands we see in canon were broken quite recently; Hagrid's wand would have been broken about fifty years ago, so he's had a whole lot more practice in trying to use it.
Furthermore, the other broken wands in canon are only poorly mended; as I recall, Harry barely even tries to jury-rig a fix for his wand, and Ron's is Spell-o-Tape'd together. Hagrid's wand, on the other hand, is effectively being held in a splint, if we assume that he put the broken pieces of his wand in the stem of his umbrella instead of in the ribs or something like that.
In fact, one could even argue that by putting the broken pieces of his wand in there, Hagrid has effectively crafted himself a new wand - after all, wands are composed of a magical core surrounded by by a wooden body, and by putting the magical core of his broken wand in the wooden stem of his umbrella Hagrid may have accidentally made something that is significantly more effective than a broken wand, yet not quite as powerful as a "real" wand.
Finally, and perhaps the largest stretch, consider the fact that wands are quasi-sentient - much like many animals, with which Hagrid has quite an affinity. His wand may simply like him enough that it's willing to put in the extra effort to help him out, and thus work better than one would expect a broken wand to.
Thus, although there doesn't seem to be much canon evidence for why he's able to do magic, it is reasonable to think that Hagrid's ability to use magic with a broken wand stems from years of casting spells with it, and his more-effective repairs to it, and perhaps even his sunny disposition.
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