Omegle tracks users using their IP address, which allows them to know where information is to be delivered. Through tracking, Omegle can link you with other users for a chat, and they can also use the same IP address to ban you from using their service by just blocking your IP address from linking to the chat servers.
Therefore to unban yourself from omelet, you would have to somehow find a way by which you can change your IP address. There are a variety of ways by which you can get this done, which includes resetting your router, contacting your Internet Service Provider so that they would assign you a new IP, unplugging your cable modem, and so on.
However, there is an easier way to go about it, which is the use of Virtual Plane Network (VPN) to get unbanned. VPN is a great way to unblock all kinds of online content and services, including Omegle.