1. You get physically reconstructed into a better, healthier, version of you.
No matter what shape you are in before you start Judo or JiuJitsu, most are going to spend the first 3-6 months feeling like they are being broken down and reconstructed. For some, this is a pleasant experience and others, a test of will. It's not uncommon for people to be sore every day after training during this period. But fear not, this is your body adjusting to tons of new muscle development in areas you have probably never used consistently before.
With continued training, your cardiovascular health and breathing naturally improves. Core strength will improve dramatically, combating a host of injuries that come with age, like back pain. The best results come from those training 3+ days per week and will also include a myriad of other benefits like increased flexibility, fat loss, and lowering of blood pressure. Grappling is one of the best overall exercises because it is both anaerobic and aerobic. Grappling works out nearly every part of the body; you could call it the "Original CrossFit". Your balance and coordination will also become significantly better, which helps in almost every aspect of everyday life. Many, even at 50+ years of age, report that after a year of training regularly they are in the best shape of their lives.
Judo can take a lot of dedication to stick with it, but the results are worth it. A common saying holds very true; "It doesn't get any easier, you just get better and tougher."
2. You become mentally stronger.
One of the most important lessons that you learn in Judo is that it pays to be calm under pressure and that hard work pays off. Because you can spar at full force with little risk of injury, you will quickly learn that the best way out of a tough situation is to stay calm and push right through it instead of quitting. It also teaches you that nothing comes for free but if you work for it the rewards are tremendous. For many, especially kids and young adults, these lessons can form the foundation of a successful and happy life.
3. You regulate stress and mood more effectively.
The other, often overlooked benefit, is the impact on mental health. Studies have shown that Judo and BJJ can provide a powerful way to combat depression, anxiety, and PTSD. A study done in 2003 at Southern California University showed a marked positive impact on symptoms for those who participated in regular Judo practice. http://judoinfo.com/research21/. Studies like this are also backed by loads of anecdotal evidence from regular practitioners of Judo. In every dojo you will find people who have had life-changing experiences though starting Judo or BJJ. Many say "it's cheaper than therapy, a lot more fun, and a lot more effective."