Why does my Apple Watch keep asking for a password?

Is it a new or used watch?

Activation Lock is designed to prevent anyone else from using an Apple Watch if it's ever lost or stolen.

About Activation Lock on your Apple Watch - Apple Support

Activation Lock is an anti-theft feature that cannot be bypassed. Only the previous owner can remove the lock by using their Apple ID & password. Apple will not help you find the previous owner. If the previous owner cannot or will not remove the lock, then the watch is useless to you or anyone else. Try to get a refund (through eBay) or dispose of the watch properly. In the future, see the 'Before you buy a watch from a third-party' section in:

What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support

If you are the ORIGINAL owner, then see the 'Start a support request' section in:

How to remove Activation Lock - Apple Support

Also see lobsterghost1's tip:

The All Too Common SAD Reality of Buying … - Apple Community