How did the French cause the British Empire to expand their colonies in the Scramble for Africa?
The French did this by trying to expand their own empire in Africa. Because the British did not want to be outcompeted by the French, they tried to expand as well. This can be seen in the Fashoda Incident.
Both France and Britain dreamed of having huge African empires. As France tried to expand its empire east from Senegal across Africa to Ethiopia, Britain felt its dreams were being threatened. It wanted a north-south empire stretching from Egypt to the Cape of Good Hope. As France expanded, Britain therefore tried to expand. This led to the confrontation at Fashoda (in what is now Sudan) in 1898.
Thus, France caused England to expand its colonies in a reaction to French attempts to expand their own.