You're going to need to complete some of the objectives set by your team each race weekend. These include qualifying and completing races above, or equal to, the position requested by your team. Doing so will increase your reputation level. As it increases, other teams will become interested in securing your services for the following year.
You can check the interest you've attracted from other teams by selecting your agent in the trailer. Also in the trailer, you can see the team standings. Eventually offers will start rolling in. Make sure the contract offer is from a higher placed team, then accept it.