Outsourcing is when you contract out your work to any other person or firm for a significant period of time. The party which gets the work is responsible for completing the work successfully.
When a company outsources its business function to another company, it does so to concentrate on other nucleus prospects of the business.
Similarly a firm which wants to outsource web development to another firm, it does so to make it more profitable than doing the work in-house. 
The reasons can be numerous. We can take three scenarios and see how businesses can profit by outsourcing to a web designing company or a web development company.
Scenario 1
Consider a web development company in a country like USA. They are approached by a customer for a project to be done using a latest technology in which they don't have the expertise. Now the firm has to either let the customer go or outsource web development to another firm who have the required expertise. That way they not only get a customer they also handle to widen their services if they come to a prosperous relation with the web firm they outsourced their work to.
Scenario 2
If a firm is very engaged and they have numerous customers they can't afford to provide services to all at a particular time, they can consider outsourcing. If they manage to find the right firm, they can successfully manage their customer projects and the customers. The key is to find the right web designing company to outsource the work to.
Scenario 3
If a firm is looking to flourish its business and services there are many things to be done. The first is expanding the base for it. The next step would be to take more employees and also advance the skills of the existing employees. All this requires additional money and if a firm is not in a position to endow more money, the best solution is to outsource the work to the right company.
If they outsource the work to another country web development company, they will not only save on these costs but will also get the work done at a fraction of cost they would be charged for in their country.
Similarly, say a firm has all the expertise to work on web development project but they do n't necessarily have the expertise in Search Engine Optimization. So if they are approached by a customer who wants web development as well as SEO services, they can do the web development part in-house and outsource the SEO work to a SEO services company.