1. It recharges you
Short Answer: It is easy to get worn out which leads to burn out and frustration. Taking even a little time for yourself can help you re-focus and give you the drive to keep going.
More details: We go-go-go all the time. We do it because not only of what is expected of us, but also because of the expectations we put on ourselves. Anything else in existence - from your phone to your car, to an old fashioned pocket watch - needs to be rewound, refueled, or recharged in order to keep working. Your body is the same way. You need to take time to stop, turn it off, and recharge to keep working at your best.
2. You can't help others if you are drained
Short answer: The old adage "you can't pour from an empty pitcher" applies here. If you are needed to give to others, provide for them, or help them, you can't effectively give what you need to give if you have no energy or anything of yourself left to give.
More details: In order to give, you have to have something to give. Whether it is doing your best in your job, interacting with others, coming up with ideas, or simply keeping everything functioning and moving, it all has to come from somewhere. Again like the example of a car or phone - if it isn't full of gas, there is nothing to allow it to run. If your phone isn't charged, it won't work. Same for your body. Your mind needs time to rest and refuel to keep producing ideas and making decisions. Your body physically needs time to recover each day before it can keep going. When we skip the stuff that we need to keep ourselves going, we don't let our bodies fully recover and allow ourselves to fill up with all we need to give back what we need to provide.
3. Lead by example for those who look up to you
Short answer: Kids, co-workers, siblings, students, or whoever you come into contact with in your life, you make an impression. That impression could be good or bad depending on how you take care of yourself. You want those around you to take care of yourself, so you should show them that it is ok.
More details: People are always looking up to you - even if you think you are not a parent, teacher, or in a significant position. Every act in your daily routine, or even your existence has an effect on those around you. Our society has become very self-centric lately, in which people only think of themselves, so it is difficult to believe that how you act or treat yourself has any relevance for anyone around you, but human beings are still social beings who learn from what they see others do. If you interact with anyone, you influence them somehow. If you realize it or not, you do set an example for those who see you or are around you. If you do know you have people who look up to you like children or co-workers, you want to be a good influence, one who encourages others to be the best versions of themselves. You can't do that if you aren't taking care of yourself. People see you and how you act, think, speak, and your attitude towards life. Nobody wants to influence someone else to be grumpy, stressed, and miserable - so always let yourself be a good example for others by showing them that self care can make you a better person, and maybe that good will rub off on those around you.
4. Builds relationships
Short answer: When you take care of yourself, it can build confidence and keep your focus sharp. This allows you to have better communication skills and makes you just generally more pleasant to be around, so it makes building connections with others more easy and fulfilling.
More details: Taking care of yourself can have some unexpected benefits. If you need to collaborate, network, or even just come up with fresh ideas, you can do that so much better when you are calm, well-rested, and have a fresh mind. Taking care of yourself clears out the cobwebs and helps you be able to communicate more clearly, come up with better ideas, and increases your ability to communicate with others, which in turn makes it easier to build connections, strengthen teams, and solve problems. If you are overly stressed, it can cloud your mind, make you cranky, and make others not necessarily want to bother you, ask you for anything, or rely on you. Calm, cool, and confident is easier to portray if you take care of yourself and let yourself become your best.
