To insist on something is to demand or swear to it. You can insist that you didn't eat the last piece of cake, and then insist that your brother show your mom the bits of frosting on his fingers.
The verb insist comes from the Latin insistere, meaning “persist,” “dwell upon,” or “stand on.” To insist on something, like a room with a view, is to say that you must have it. To insist that you never met that man before in your life — even though there's a picture of the two of you together — is to claim that you're telling the truth. To insist on tapping your foot on the chair leg is annoying.