What does "Can it!" mean, as used by Alex Chadwick?

The original, literal sense is “to preserve (food) in a can”. Derived from this meaning is the secondary meaning of “stuffing away” or even “firing (from a job)”.

This particular phrase is a highly idiomatic one meaning, “Shut up!”—basically, you’re saying, “Take whatever it is you’re saying, put it into a can, and seal/preserve it there, out of my earshot!”

Oxford Dictionaries definition

I hate to say this, since I prefer decorous language to gutter lingo, but "Can it!" is a shortened and more decorous way to say "Shit can it!" or "Take what you're saying and throw it into the shit can!" because that's all it's worth. It has absolutely nothing to do with food preservation.

You know what SNAFU and FUBAR stand for, right? "Situation Normal: All Fouled Up" and "Fouled Up Beyond All Recognition". Except that the "F" actually originally stood for some other word, a word which we avoid using in polite conversation.

Same deal with "Can It!"