on their smtp (outbound) mail servers. This is to prevent people using their servers to send millions of spam messages. It enables the server to check that only people with valid identities are sending.
They will also prevent you from relaying, which means you can't use their servers to relay messages to another smtp server. When you send via the Waitrose site you are using a Waitrose server,your message isn't passing through the TalkTalk smtp server.
Alternatively, just set your email programme to authenticate on the outbound server - enter the username and password that TalkTalk supplied.