How do I retrieve my Messages data from an old iPhone to my new iPhone without manually doing it ?

Unless you have a backup of the broken phone you can restore your new phone to or you were using iCloud Syncing for Messages while using the broken phone (sounds as if you weren't), then no, the messages are gone.

It's important to backup phones regularly, preferably daily, so that, in the event something bad happens to the phone, you don't lose much data. If you use iCloud to back up your data, all you need to do is plug the phone in every night, make sure it's connected to WiFi and lock the screen. It will automatically back up. Then, should your phone get broken or stolen, you won't have lost more than a days worth of data.

See here:

How to back up your iPhone or iPad with iCloud - Apple Support

Set up iCloud for Messages on all your devices - Apple Support

You will probably need to do a factory reset on the new phone and then restore it with the most recent backup you had from the broken phone. Make sure not to pick a backup from the new phone, you want to make sure that the backup you choose was one that happened when you had the broken phone on.

ellisia65 wrote:

Can you explain to me how to put my backup information on my new phone because it currently says that my iCloud backups are on, I never turn it off. (I also have icloud+)

The instructions for how to back up your iPhone are in the first link I provided in my previous post.

If you do not have a backup from the broken phone, you can't restore that information to your new phone. If you do have a back, see here:

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

Note that restoring the phone from a backup will overwrite everything currently on it.