It means to be obvious about your feelings or being unable to hide your emotions.

The earliest recorded use of it recorded is in Shakespeare’s “Othello,” said by a character named Iago:

“But seeming so, for my peculiar end:
For when my outward action doth demonstrate
The native act and figure of my heart
In compliment extern, ’tis not long after
But I will wear my heart upon my sleeve
For daws to peck at: I am not what I am.”

It’s likely a reference to the old medieval culture. Prominent families have their representing colors, and a knight would wear a ribbon in that color to express his fondness to a lady in that family – thus, wearing his heart on his sleeve.