To get anywhere in life, you must follow through your actions. Hesitating or not following through can allow success and chances to slip away. Of course, following through is a big part of combat sports such as MMA and boxing. Following through can be the difference between a stoppage and having little effect.
When it comes to following through in striking, it can generally mean two things. Either follow through a combination by throwing another punch which can have devastating effect, or trying to somewhat punch through the target. Trying to punch through target makes the punch much more piercing and ensures that all the force gets transferred into the target and more. This can have a much more powerful effect than pulling it back or letting your punch go limp.