Legal question. Why do we need a "Notice of Motion" to move the court?

Where did the Notice of Motion come from? What is the history behind such a form?
Answer:
The Notice of Motion is required to give all parties to the case, and the Court, fair notice (or warning) that a party is requesting that the Court do something. Its purpose is also to start the clock running on the time limit for the other party or parties to respond if they want to oppose the motion.

The history is somewhat different in each jurisdiction, and is enshrined in the Court rules (or Rules of Civil Procedure) that are used by that Court.

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Any request to the court is a motion.

You are moving (asking) the court to do something. That terminology derived (like much of American law) from old English legal practices centuries ago.

As part of any motion, you need to notify the court and the other party that you are filing a motion. That's the Notice of Motion. It basically repeats the header for what you are asking, and informs the other party when the motion hearing is going to be held.

It's one of several dozen different procedural aspects involved in the litigation process. Failing to follow the proper procedures can get the motion thrown out entirely.