We want to use phpmailer https://github.com/PHPMailer/PHPMailer on our Website. I read all I could find on GitHub and with Google but am unable to understand how to give proper credit for using phpmailer on our Website.
Does it has to be visible on the Website or could it be a comment in the hmtl-sourcecode?
How should the wording be?
You're not required to give any credit, though of course it's nice if you do - a simple "We use PHPMailer" and a link to the github project would be appropriate. The same applies to any other open-source projects you use.
The main restriction of the LGPL 2.1 license that PHPMailer uses is that if you make any changes to the library itself (which you don't usually need to do), those changes inherit the LGPL license and the source should be made available on demand. Note that that doesn't mean that you need to release your app's source code, or your code that uses PHPMailer - that's a requirement of the full GPL license, which PHPMailer doesn't use; this is the main distinction between the GPL and LGPL licenses.
Of course another way you can support open source projects is by donating / sponsoring them and their maintainers either through GitHub, Patreon and similar sites, or if you're a company, consider Tidelift.