"Go on" often means to continue or to proceed.
"The meeting went on for two hours."
"I've talked enough about it already, I won't go on."
"The team lost its first match, but went on to win three in a row."
When the two words are used alone in a command like this, it is just a way of encouraging somebody to do some action. Depending on the context and the tone of voice, it can be a friendly way of giving permission or making a suggestion, or it can be an angry command.
:-) "Go on, have a biscuit!" (= "You are allowed to eat a biscuit", "Please eat one of these biscuits.")
>:-O "Go on, get me a biscuit!" (= "I'm hungry, damnit, get me a biscuit!")