How true is this verse! The thought of a dog throwing up is not a pleasing one... And if you've ever seen it happen - it's not pretty to see either. Well, the thought of sin should bring the same kind of reaction as a dog throwing up... It's disgusting.
But how often do we sin or do something that we know is wrong, feel really guilty afterward, and then let ourselves do the sin again a few days/weeks/hours later? That's exactly what this verse is talking about. When we repent of our sinful actions, we need to commit to that and fully repent - and not return to that sin.
So let's not be like the dog that returns to his vomit... if/when we sin, let us see how horrible and disgusting that is to God - and turn completely away from that.