Why is continuing the swing after it hits important? Every time a part of swing is stopped or slowed prematurely, it loses power. Even adjusting the angle of attack can reduce the power of the swing. Changing the speed or direction of a swing forces your body to do extra work. This combination means that a swing that you "put a lot of effort into" may end up still hitting light. In turn, that causes your perception of an opponent's call of "light" be tinted by your own inflated feeling of how hard you swung. So not only are you hitting way lighter than the effort put into the swing would suggest, you are also building up a feeling that your opponent has a high hit tolerance.
As I've mentioned in Overswinging, the bounce of a shot helps aid in recovering or comboing it. By not following through, you fight against the force used to swing the shot in the first place, and get rid of some of that bounce. By not following through, you are essentially wasting energy, either by swinging extra hard to compensate for the wasted effort to get a sufficient hit, or by swinging a wasted shot that will end up being light. Through a long day of fighting, that wasted energy will take its toll.
You will notice above, I mention that the body must continue its motion as part of the follow through. This is important, as some out there don't explain that simply letting the sword motion continue isn't the same as following through with the swing. From the moment your body stops the motion, the shot loses power.
From the moment your body stops the motion, the shot loses power.
Lets look at an example. A common situation where I see this regularly is with min reds. As I've mentioned before, they aren't really suited for breaking shields efficiently. This is made even more true when the user doesn't follow through with their swings. Many times I have witnessed a min red user attempting to place many shield shots in rapid succession, only to have them all called light. The problem is that by trying to fit in as many swings as possible, the user ends up having to stop their body's motion before the shot has connected. Essentially, their body ends up starting the motion for the second swing by the time the first has hit. Not only are they reducing power by not following through, they may be further hindering the hit further by pulling their arms back to prepare for the second swing before they've even hit.
In order to deliver a good, solid shield hit here, the min red needs to keep their body motion until as much of the force of the swing is transferred to the target as possible. Attempting to block a counterswing or recover to guard are often causes of not following through. This means, for min reds, that often the choice is between being able to block, or delivering enough force to be a solid hit.
Following through doesn't only apply to slashing weapons. Stabs have a stark contrast between a good hit and a light hit. Without any follow through, a stab might touch the target, but it won't always be sufficient. Even with only a short bit of follow through, however, the force goes up drastically. Stabs, then, should be aimed to be stabbed "through" the target, not stop on them.
A common cause for stabs landing light from not following through is actually range. At max range, the body can no longer continue the stabbing motion, which causes it to greatly reduce power as it approaches this range (as different parts of the body reach their limits of extension or twist). Roughly, the last few inches of reach that can touch a target will have a significantly lower impact than a shorter ranged stab. Those "last few inches" turns out to be about how far of a follow through is required to land a solid hit. By reducing that follow through, you reduce the hit force proportionately.
One thing to note here, is that following through also might result in excessive hits in some cases. For example, a close/medium range stab with full follow through, to an opponent that is unaware, will likely hit on the side of excessive. The same is true for backstabbing with a sword. By choosing specifically not to follow through with the entire body, one can scale back their hit force for their opponent's safety and generally courtesy.