King James VI of Scotland succeeded Queen Elizabeth I of England, effectively uniting the two kingdoms. In a nifty symbolic move, James also chose to combine the royal arms into that of one united kingdom. The Scottish coat of arms had used two unicorns supporting a shield, and the English coat frequently depicted a lion in this regard.
Therefore, James decided to use both national animals in the new coat of arms, which was an especially powerful gesture considering that lions and unicorns were viewed to have been mortal enemies.
(Sources: 1, 2)