To understand how each party would benefit from this type of arrangement, consider a situation where each party has a comparative advantage to take out a loan at a certain rate and currency. For example, Company A can take out a loan with a one-year term in the U.S. for a fixed rate of 8% and a floating rate of Libor + 1% (which is comparatively cheaper, but they would prefer a fixed rate). On the other hand, Company B can obtain a loan on a one-year term for a fixed rate of 6%, or a floating rate of Libor +3%, consequently, they'd prefer a floating rate.
Through an interest rate swap, each party can swap its interest rate with the other to obtain its preferred interest rate
Note that swap transactions are often facilitated by a swap dealer, who will act as the required counterparty for a fee.