The phrase "to engage someone's services" means to hire or employ a person for their skills, expertise and/or labor. It implies that an agreement has been reached between two parties - the individual providing the service (the employee) and the party requesting the service (the employer). The term can also refer to using the services of someone on a temporary basis, such as hiring a contractor or freelancer for a specific project. In essence, "to engage one's services" means to make use of an individual's abilities and talents in exchange for compensation or payment.