First, make sure VoiceMeeter Banana or Potato is set up and working properly. To do this, go to Settings in VoiceMeeter and Under Virtual Inputs, make sure B1 is turned on for AUX.
Then, open Discord and click the friend icon ( ) next to someone you want to talk to on Omegle. In the dialog that pops up, type /me into the chat box and press enter. Type your Omegle username in the corresponding box and press enter. Your friend will then be able to talk to you through Discord!