You will be paying them a premium to take those responsibilities, and also they may replace your employees with less capable workers. You will be basically contracting your headcount from this agency rather than them being your employees. Any gripes with the workers will have to go through them.
If you are dealing with technology and engineering, and hand your engineers over to this company, now the engineers will be required by laws in most states to have professional engineering licenses to offer you services rather than doing work in house. Also the company offering the services will need to be licensed with the state board of engineers, or if another engineer reports them, they can and will be shut down leaving your company in a world of hurt. (the thing is the engineering boards have been looking the other way at these arrangements, but eventually they will be reprimanded.)
I am sure other industries will be similiar. Farming out the lower end jobs may be okay, and make since when it comes to downsizing or ramping up headcount for short term bursts.
If it sounds too good to be true, it probably is.