Inevitably, we all have to take time off. Whether it's from illness, work getting busy, vacation or holiday travel, or when we're just too sore for too long, no one should be dragging themselves to the gym year round with no breaks.
While it is definitely possible to continue training through a competition (and depending on the lifter's experience, we may even plan on having lifters train through a meet or two to build up to a bigger competition), it's usually not the best way to handle your body and its ability to recover. Your body has a finite amount of recovery capacity, and the training for and the actual day of competition takes up a lot of that demand on recovery. Jumping back into training a few days after a competition means that when you start a new training cycle, you're in a pre-fatigued state, and you may not progress through each phase of the cycle as expected and make planned rep PRs. Taking a week off after a competition is a good way to "reset" your fatigue from training, and while you would lose a little strength and conditioning from a week break, it would be quick to build back up through regular training, and the harder days deeper into the training cycle would be easier to tackle having started with a fresh slate.
Ilya Ilin, former 2008 and 2012 Olympic Champion and breaker of several world records, was known to take six months off after the Olympic Games. Even after a long lay off after 2012, he broke the clean and jerk and total records at 105 twice at the 2014 World Championships in Houston and again at the President's Cup in Grozny in 2015. The long breaks from lifting didn't make him weaker, it allowed him to recover from the years of hard training preceding the break, and also start the next Olympic cycle of 4 years fresh and ready to break more records.