Basically an employer contracts with a PEO, professional employer organization, who provides the employer with workers for a fee. Think of temporary employees or temps, the process is very similar with the exception that employee leasing provides for a type of permanacy where temps are by nature temporary.
You may find where some leasing companies actually hire the employers existing employees and then lease them back to the employer. The PEO's then handle all payroll taxes, reports and provide employee benefits and workers compensation coverage for the leased employees.