In order to understand the concept and benefits of an Internet Exchange (IX or sometimes IXP - Internet Exchange Point), understanding the underlying concept of the Internet is very helpful.
There is no single organization that is the unique and ultimate source of the Internet. The Internet is built by every carrier, ISP and network operator offering Internet access – hence why it is often referred to as 'the network of networks'.
In order for the Internet to work in the manner we are all used to, all those individual carriers, ISPs and network operators need to exchange data, or, put it another way, grant others access to their network. That access is either agreed on a payment basis – for transit/upstream – or on a cost neutral basis, otherwise known as peering.
In order for peering to work efficiently, Internet Exchanges offer a neutral local network where any carrier, ISP or network operator can connect and exchange traffic. The IX uses switching equipment to build the local network, by placing equipment in existing carrier neutral data centers - sometimes in more than one location for redundancy purposes, and uses dark fiber to interconnect those locations.
Euro-IX, an association of Internet Exchange Points, made a video explaining IXPs: "Internet Revealed":