The first lion came from Henry I – known as the lion of England – who had a lion on his standard = one lion on taking power in 1100.
He married Adeliza, whose father also had a lion on his shield, and to commemorate the event he added a second lion to his standard = two lions.
In 1154, two lions became three lions when Henry II married Eleanor of Aquitaine, who – yes – also had a lion her family crest.
So three lions.
Later in the century, Richard the Lionheart used the three golden lions on a red background as a symbol of the English throne and, after that it appeared on the Royal Arms of every succeeding monarch.
So when the FA was formed in 1863, they based their logo on this royal shield of three lions.
Since then, the design has only been changed once, and that was in 1949 – where the crown on top of the lions was removed to differentiate the badge from the English cricket team.