Folks, a smithy is the place where a smith works, a forge. Today it seems the confusion is so rampant that some recent dictionaries are even listing "smith" as an alternate definition of "smithy" - aaarrrrrrrrghhhh!
Please don't torture me further. When all else fails, remember the first stanza of Longfellow's poem and picture the "smith, a mighty man" working at his forge ("smithy") under a spreading tree -
Here endeth the lesson.Under a spreading chestnut-tree
The village smithy stands;
The smith, a mighty man is he,
With large and sinewy hands,
And the muscles of his brawny arms
Are strong as iron bands.
Go and sin no more!