The DNSSEC (Domain Name System Security Extensions) chain of trust is a verified electronic signature, or handshake, at each DNS lookup node. In other words, it is a chain of lookups validated by the domain name's digital signature that secures the request through all lookup nodes. This ensures that no rogue or illicit player can slip into the lookup path and redirect the lookup to a bogus site.
Here's an example of using your browser to visit coolexample.org:
From your local computer to the authoritative nameserver for the requested URL and back, a digital signature (or handshake) at each node insures that your request provides the website you requested and that the request is not intercepted by rogue operators along the way.