If your mother says she will defray the cost of your next move, say thank you. She means she will take on some of the expenses for you.
Your school band might hold a bake sale fundraiser to defray the cost of new band uniforms. When they do this, the amount raised will be taken off the total cost of the uniforms so that each individual band member has to pay less. If you live in New York and are invited for a job interview in California, the company may defray the expenses of your plane ticket and hotel, especially if they really want to hire you.