Many people underestimate the importance of "the extra rep". This was always a big concern of mine because if you really want to get fast results, getting the extra rep is really the only thing that can trigger it.
What is the extra rep? It's not the rep where you "feel the burn", or even begin to feel the burn, or the point at which you "think" you can't get another rep.
It's the point where you feel and think that you cannot get any more reps and you would usually stop there, but with "good form" (if not perfect form and total concentration) you push and force yourself to get 1-4 or more reps.
For the sake of the mental component, the rapid results triggered by these extra reps are critical because if you don't start seeing dramatic results and transformation of the body and muscle, you will likely lose interest, and progress will either cease all together and lead to a plateau, or lead you to return to where you started with your original physique.
When you get the extra reps, you actually cause a biological transformation to take place; you force the muscle to call on more muscle fiber to help and assist lifting the weight. This is because all of your other muscle fiber is "exhausted", and this new muscle fiber is usually not utilized when you don't get these extra reps.
When you now utilize these usually unused fibers, they, like all muscle fiber, have to respond, and this in turn causes them also to exhaust, or "break down".
I refer to this as "muscle fiber recruitment." The more muscle fibers you recruit (and the best way to do this is by getting the extra rep), the stronger, bigger, and more defined (depending on the weight you use will determine which of these) your muscles will become.
The reason why many people do not achieve the extra rep, along with psychological reasons, is because they begin to feel the pain in the muscle that is sometimes caused by lactic acid buildup. This pain is associated with injury, but in the process of body sculpting and bodybuilding, in a sense, injuring the muscle is exactly what you need to do to stimulate the best results.
This sounds harsh, but "breaking down the muscle fibers" is, in a way, an injury to the muscle. The fibers tear during the extra reps and this is what allows the protein and nutrition to enter to repair and rebuild them stronger afterward. But the only fast way to ensure they tear and break down sufficiently is to get the extra rep so that you can trigger the results right away.
You must work through the lactic acid buildup, ideally with excellent form, and the muscle fiber "burn" and force that extra rep to truly trigger the natural muscle, strength, endurance, definition, tone, or size gains (depending on what your goals are).
Always try one more time to get the extra rep. Like anything in life worth achieving, failure must be reached before you can progress. Failure of the muscle leads to growth. If you can't take the set to true muscle failure, then you'll never grow, and you'll stay in a place of "stagnation", or plateau.
This is why the extra rep matters and how important it is. If you want to succeed, you must get it, and you must never give up until you do.