Newer browsers protect you better against scams, viruses, trojans, phishing and other threats. They also fix security holes in your current browser!
Every new browser generation improves speed
Websites using new technology will be displayed more correctly
With new features, extensions and better customisability, you will have a more comfortable web-experience
If you can't change your browser because of compatibility issues, think about installing a second browser for browsing and keep the old one for the compatibility.
If you are on a computer that is maintained by an admin and you cannot install a new browser, ask your admin about it.