2016 is an election year in the United States, which means that Americans will be choosing their next President. There are some candidates who are well qualified and have good, solid ideas for improving the county. In contrast, there are other candidates who don't appear to be serious and are experts at running off at the mouth; their ideas are unrealistic and many of their statements are offensive.
Parent: You must listen to me, Bobby! It's very important that you pay attention in class when the teacher is talking. You have been in trouble 3 times this week for interrupting her lesson! What is your explanation?!
Child: My teacher is so boring! All she does is run off at the mouth. No one listens to her because no one likes her.
Parent: I can't believe you would be so disrespectful!
To run off at the mouth means that you are talking a lot, but not saying anything important, or to talk a lot without consideration of your words. This expression has negative connotations and may be considered rude. Please use with caution!
Example 1 mentions presidential candidates who talk a lot, but doesn't provide anything useful and are sometimes offensive.
Example 2 shows a conversation between a parent and child. The child says his teacher just talks a lot and doesn't say anything useful, that's why he is having trouble in this class. His mother is surprised that he would be so rude.