William became king of England in October 1066 after defeating Harold at the Battle of Hastings.
To understand why he brought an army over from Normandy, you need to know that he thought he had a claim on the throne. This is the beginning of that claim:
Many years earlier, when Edward the Confessor (Duke William’s cousin) was in exile, and before he was king of England, he promised the abbot of Fecamp Abbey in Normandy that it could own lands on the south coast of England, in gratitude for being allowed to stay in Normandy. Lands near Hastings had also been given by a former English king, Cnut.
This is partly what gave the Normans claim to lands in England, but the Saxons sent the monks packing back to Normandy in 1052 because they didn’t want any Normans on their lands.
The Saxons who did this were led by the Godwinsons. But William of Normandy could use this to his advantage, claiming that he could get the abbey’s rightful lands back. That also put him in favour with the Pope (although the Pope’s price included more land if William won).
When it came to the time of the invasion, the abbey even paid for one of the ships William used.
You can find out about how the battle went and what happened after in the video and book below.