In a nutshell, here is some of the benefits of running a witness server include:

• more understanding of how Hive works on a technical aspect
• important contribution to the community by participating in signing transactions and holding a copy of the blockchain
• have the opportunity to have a say in the decision on what rules are getting implemented in the blockchain
• if you receive enough witness votes you can start earning enough to pay off your server rental and maybe even make profit

To run a witness:

• you need to have some technical skills or partner with someone who have them
• rent a server or purchase one and make sure it will have 24/7 stable internet access
• compile, install, run and maintain the HIVE blockchain app
• starting a witness vote gathering campaign
  • you only get paid if you sign blocks
  • you only get blocks to sign if you have received some witness blocks
  • the more witness votes you have received the more blocks you get to sign

There are several guides out there but here is a recent one:
https://hive.blog/hive-139531/@balaz/step-by-step-guide-t-28-03-2023-qn2jd