One of the meanings of "go on" is to continue talking about something. For example, if someone stops talking and you want to encourage them to say more about a topic, you say:
When you use "go on for ___" or "go on about ___", it expresses that someone is talking for too long. For example, in the sentence:
She went on and on about her dog.
...it's clear that the speaker didn't want to hear about the subject's dog.
You can use "go on for ___ about ___" to communicate both the length of time and the topic, or you can use "for" or "about" by themselves:
She went on for 15 minutes after you left.
Every time I talk to him, he goes on about why I need to move back out to Arizona.