An example of insist is when you demand a new plate of eggs and won't take no for an answer.
An example of insist is when you follow through with your plan to go on a picnic, even when it is raining.
An example of insist is when you say you feel fine and keep saying it over and over without waivering, even when people question you or doubt the truth.
We insist that you stay for dinner.
I insist on paying my share of the expenses.