What does "shoot somebody a smile" mean?

To shoot:

to direct suddenly or swiftly:

• Shoot the spotlight on the doorway. He shot a smile at his wife.

(Dictionary.com)

to cast

a look/glance/smile [transitive] cast (somebody) something to look, smile, etc. in a particular direction. "She cast a welcoming smile in his direction."

It is similar to the phrase "flash a smile" which compares a smile to a photograph or beam of light that triggers positive response from the target, except “shoot somebody a smile” includes an aiming component that will sometimes imply the ability to stun/paralyze/["smite"-slang] the target.

The association of ideas is with gunfire.

If a firearm is discharged, the firing of a single shot is an extremely brief event. Shooting a smile at someone is a way of saying that the smile has an equally brief duration -- and often is used to suggest that it may have also an equally great impact (although not in a literal sense).