Grinning broadly.

Origin

The origin of this is uncertain. Of course, we know the phrase because of Lewis Carroll's Alice's Adventures in Wonderland, (published 1865) and John Tenniel's illustrations in it:

'Please would you tell me,' said Alice, a little timidly, for she was not quite sure whether it was good manners for her to speak first, 'why your cat grins like that?'

'It's a Cheshire cat,' said the Duchess, 'and that's why. Pig!'

She said the last word with such sudden violence that Alice quite jumped; but she saw in another moment that it was addressed to the baby, and not to her, so she took courage, and went on again:

'I didn't know that Cheshire cats always grinned; in fact, I didn't know that cats COULD grin.'

'They all can,' said the Duchess; 'and most of 'em do.'

We do know that Lewis Carroll (The Reverend Charles Lutwidge Dodgson) didn't coin the phrase himself, as there are citations of it that pre-date his stories. John Wolcot, the poet and satirist, who wrote under the pseudonym of Peter Pindar, included it in his Works, published variously between 1770 and 1819 - "Lo! like a Cheshire cat our court will grin".

William Makepeace Thackeray also used the description well before Dodgson, in The Newcomes; memoirs of a most respectable family, 1854–55:

Mr. Newcome says to Mr. Pendennis in his droll, humorous way, "That woman grins like a Cheshire cat."

There's no convincing explanation of why Cheshire cats were imagined to grin. It seems likely that no one really believed that they actually did. We can take the next line in Thackeray's piece - "Who was the naturalist who first discovered that peculiarity of the cats in Cheshire?", to be sarcastic.

The numerous folk-etymology derivations that explain how Lewis Carroll came up with the idea have to be spurious, as we know he didn't. Alice's Adventures in Wonderland has a long troupe of fantastical animals. It's very likely that Dodgson had heard of Cheshire cats being said to grin and adapted the idea into his story.