Someone else holds the domain in their name on your behalf. You control it; they're the registered owner on paper.
Proxy registration exists because WHOIS used to be public by default. Want your name off the internet? Pay a registrar or proxy service to put theirs there instead. The registrar holds the keys; you get a contract saying it's yours.
Not the same as WHOIS privacy. Privacy redacts your details but keeps "privacy service" in the record. Proxy registration swaps you out entirely—the proxy is the registrant.
Why it matters: harder to trace ownership. Useful for journalists, activists, people in hostile jurisdictions, anyone who doesn't want their name connected to a domain. Also useful for people running multiple sites who don't want a pattern visible.
Downside: you depend on the proxy not disappearing, selling the domain, or complying with a court order. You're trusting a third party with an asset. Some registrars won't allow it. Some TLDs ban it outright.
GDPR and CCPA have made WHOIS privacy the norm in many places, which reduces the need for proxy registration. But proxy registration still serves people who need deeper obscurity or operate in jurisdictions where privacy shields don't apply.