I don't want to hear the story from the beginning. Cut to the quick and give me the scoop.
A: Quick is actually defined as meaning flesh or skin.
That definition has survived in reference to the fingernail. The quick is the part of finger, beneath the nail, that relies on the nail for protection. If you cut your fingernail too short and (painfully) expose the tender flesh beneath, you have cut it to the quick.
In hand to hand duel or battle a combatant who wished to taunt his opponent, might intentionally cut through the clothes or armor to the quick. The act of cutting through clothes to the flesh is to cut through the insignificant to the substantial.
If someone cuts you with a knife but it's a shallow wound and inconsequential, then you wouldn't use this phrase. If they cut you deeply, or stabbed you, then you might say they'd cut you to the quick. Again this is cutting though the inconsequential to the meaningful.
This may also be related to the use of the word quickening to describe the hypothetical moment when a fetus becomes a baby. This term and idea were once very common in both Anglophile Europe and the Americas.
In either case, there is a clear meaning of penetrating the dead, dull stuff and getting to the live, important bits. This translates very nicely to the most common use of the phrase today, to mean skipping to the important part of a story or explanation.
Alternative: Cut to the quick has the same origin as Quick and the Dead. Here quick comes from the old English Cwicu meaning living and thus to cut to the quick implies a deep wound into living flesh.