This is a topic I see come up a lot, and while they do offer an export option within Local I do prefer to do it the manual way as I can make sure everything I want to be copied over to the new host is copied. All of these steps were done on macOS with Local version 5.6.3+4270, so depending on your operating system and version of Local this may be different.
The first thing I do is right click on the site I want to export to a new host, and choose Reveal in Finder as shown below.
You should now be presented with the files in the finder where you can see the location of /Users/shelby/Local Sites/localtest/. Now that you know the location you can use your FTP client of choice to upload them to your new host, hopefully via SFTP.
Head back over to Local and click on Database for the site you want to export, and then click Open Adminer.
Adminer will open in your web browser, and on the left hand column of the page you will want to click on Export.
On the export screen choose gzip, and then SQL for the format, now click Export. Your browser should download a copy of the exported database that you can import on the new host.
Again the normal export option that they provide when right-clicking on the site will likely work for everyone, but I did want to provide a more manual walkthrough on how to go about doing this incase they remove the export feature in a future release of the software.