You will never leave the gym, finish your run or workout in a fitter state. Yes, you may feel invigorated and have some nice muscle pump, but the exercise will have caused trauma to your muscles and it will have depleted your energy stores. It is when you rest you will recover and get fitter, and so your recovery should be a cornerstone of your training programme, not an afterthought.
It is advised that you should aim for 7-9 hours sleep a night in order to really reap the rewards from your training. If you're consistently getting less then this figure, then you can expect to work a lot harder or longer for the same transformations to your fitness you'd have gotten if you slept for this duration. If you're dramatically under this figure often whilst training hard then you can expect to be heading for a nice bout of fatigue too.
Whilst we sleep the pituitary gland releases Human Growth Hormone and Protein Synthesis occurs. In short, your body is recovering and developing - meaning that when you wake you will be that bit fitter and (potentially) ready to exercise again. Some athletes will even plan in naps to ensure they maximise on this recovery period.
There is tonnes of information on the internet on how to get the best nights sleep you can so have a Google about. Key things include limiting screen time before bed, keeping the bedroom cool, sleeping on a comfy mattress and also having a routine.
When you break down the muscle, there is a muscle building synthesis that lasts about two to three days. It is for this reason that should you say do a huge chest workout, it's not advised to target this region for the next 24-48 hours. The muscles are already shredded, they need time to repair, and they will be growing. You can use the following day to target another muscle group or style of training (e.g Cardio). However, you should allow your body at least one complete rest day a week - and by that I mean you don't do any form of exercise or rehab - just let your body be. For those that love to exercise, it is often on these days you feel like a Walrus by the end of it, however be rest assured that you're body is enjoying the time off and it will only come back stronger.
For those newer to training, taking 2-4 rest days a week whilst you build up the training volume is no bad thing. We don't want to completely total the body in the early stages as this will only increase your chances of dropping out of the training plan as you feel lethargic and achy all the time.
There is this concept of 'active recovery' whereby after a hard training day(s) you will do some stretching, yoga or very light cardio. Remember, this is still being active and will require adaption from your body, and so this does not technically count as a complete rest day!
The best total rest day would be one where you do literally nothing but relax, be calm (stress inhibits recovery) and can eat plenty. As a parent of two young kids I know how impossible it would be to plan such a day into a week for those in a similar position, but are there other hours of the week you can steal for this time instead?
Every few weeks you should also allow yourself what we call a deload week. You cut your training down dramatically from the volume you've worked up to. You're training enough to keep you mentally engaged and to keep the body feeling fresh, but you're allowing the time for rest, recovery and adaption. Again, you will only come back stronger and fitter from allowing your body this time.
Sleep is not the single 'golden bullet' to recovery however. You must build it into your recovery plan along with things like stretching and cooling down immediately after your workout, and of course your nutrition. A lack of (the right) nutrition within your training week will dramatically overshadow the effects of getting enough sleep.
Happy napping, snacking and chilling!