Legal outsourcing is when a company or law firm decides to get help from another company in a different country to do some of their work. Just like when your mom asks your older sibling or a neighbor to help her with some cleaning at home, the company or law firm asks another company to help them with some of their legal work.
This other company is usually located in a different country where the people working for it are paid less and can do the work much cheaper. For example, if a US law firm needs help with research or document preparation, they might hire a legal outsourcing company in India, where labor costs are much lower.
By doing this, the US law firm saves money and can offer its services at a lower price to its clients. The Indian outsourcing company benefits, too, because it gets paid for doing the work and can create jobs for its own citizens.
It's like trading or sharing, where both parties benefit!