Symbolically, the Unicorn represents Laura. Once she has her romantic interlude with Jim, even though it doesn't work out, she feels changed, like the unicorn.
Technically, Laura becomes normal after she has been kissed by Jim, and like the unicorn, who now looks like a regular horse, can fit in with the other horses. Laura has passed a threshold in life, she is changed by her experience with Jim.
"LAURA: Haven’t you noticed the single horn on his forehead?
JIM: A unicorn, huh? —aren’t they extinct in the modern world?
LAURA: I know!
JIM: Poor little fellow, he must feel sort of lonesome.
JIM: Aw, aw, aw. Is it broken?
LAURA: Now it is just like all the other horses.
JIM: It’s lost its—
LAURA: Horn! It doesn’t matter. . . . [smiling] I’ll just imagine he had an operation. The horn was removed to make him feel less—freakish!" (pg. 86-87, Williams)
Laura feels that while losing Jim, she's loosing the last remain of fantasy. Unicorns in general representing fantasies, and in parallel, the broken one represents the lost dream.
There is also a suggestion in which Laura realizes that there's some romance going on between Jim and Tom. When giving him the unicorn, representing her broken heart, she symbolically gives Tom away, asking Jim to take best care of him.