WhatsApp does not let you send anonymous messages. When you look for this option in any application menu, you will see that it just does not exist. To use this feature, it is necessary to use an alternative solution.
There was a time when it became common practice to use a web service known as Wassame, which theoretically lets you send messages anonymously to WhatsApp numbers. However, that service no longer works. By sending an anonymous message with that platform, all we get is to share a third party's number with a website which purpose we do not know.
Another option, although it does not guarantee anonymity, but does let you use a phone number that is unknown to the receiver, is to get a second SIM card and copy the phone number you want to send WhatsApp messages to, and take advantage of the fact that the other person does not know your second phone number.
If a second SIM card seems too much trouble, you can always try applications such as eSIM Number to get a second virtual phone number in your device. However, these numbers are not free and are subject to a subscription model. You should note that not all numbers offered by these services are mobile (some only offer landlines), so you will have to do some research before deciding on one.
The closest you can come to communicating anonymously with another person is using other messaging and chat applications that let you communicate with a User ID without sharing a phone number or email. Among the most popular ones, you will find Telegram or LINE.
In Telegram, there is a bot that lets users send messages without revealing their identity, but it works more like a Chatroulette, rather than a way to send messages to your contacts:Telegram incognito bot
As you can see, the solution is by no means ideal. Most applications that advertise themselves as anonymous messaging solutions, such as Psst or Whisper, work similarly to this bot. At the moment, the possibility of sending messages to our contacts anonymously (either with WhatsApp or with any other application) is near to zero.