If you insist on something, or if you insist that something must be done, you say that it MUST be true, or it MUST happen. If you're being insistent, then you're being pushy and loud, trying to get your way and change people's minds, maybe even saying or shouting what you want over and over.
When you insist on something, people are usually telling you no or arguing with you. You might insist on playing loud music in the family room, even though your siblings insist that you turn it down.
In my family, we had a beautiful Himalayan cat named Meri. I insisted that Meri's eyes were green, but my sister insisted that Meri's eyes were blue. Who was right? Probably my sister! But we would annoy each other with our insistence on being right.
But insistence isn't always rude. It can be polite. Sometimes, when we want to give a gift or help someone, but the other person says, "Oh no, I can't take that!" or "Oh, no, I can't let you do that," then we say, "I insist." Meaning, "I think you MUST take this gift!" Or, "I think you MUST let me help you." That's often very nice and polite. Here, Homer insists on walking Flanders home. Flanders probably said, "Oh, no, you don't need to do that for me." But Homer said, "I insist."