It's a good solution when the expertise to perform some necessary function isn't present within a company. It's quite difficult for many small businesses to maintain a full time accountant and someone to look after the payroll. That person needs to be paid and isn't directly generating income, so there is a significant drain in resources there. Distribution, if it's part of the business, is almost always outsourced to some degree. Paying a courier company or even Royal Mail to deliver your goods is really outsourcing- the concept is nothing new.
Of course, you do have to pay for payroll services and other outsourced work, but you'll probably be paying a company that specialises in the field (or you should be). Like many things, it's cheaper for a big specialist to do payroll services for your company along with many others than it is to do it yourself because costs are shared.
Imagine trying to do UK wide deliveries yourself. Royal Mail can deliver packages very cheaply, because transport and distribution costs are shared by tens of thousands of letters and packages every day. You don't have to deliver everything yourself, because someone else has the infrastructure and staff in place to do it for you at good price. Payroll outsourcing works in the same way.