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.
The link you provide states clearly (check the info in the header of the page you linked):
Conditions
If you provide a link to a library which already installed on the server under LGPL then there is no problem at all as it would be dynamically linked. But if you provide PHPMailer in your distribution package as a "vendor" library you need to add a section to your documentation which clearly states and describes on how to replace this library with your own one or a modified one without losing the overall functionality of the script unless it is licensed under LGPL like the library itself, then this would not be needed. As mentioned already by Synchro all changes you made on PHP mailer inherit the LGPL license and also have to be mentioned as changes you did to the original code.
read point 2d) of the license!
d) If a facility in the modified Library refers to a function or a table of data to be supplied by an application program that uses the facility, other than as an argument passed when the facility is invoked, then you must make a good faith effort to ensure that, in the event an application does not supply such function or table, the facility still operates, and performs whatever part of its purpose remains meaningful.
In case of PHPMailer that is perhaps easily possible as you wo't need oh-mailer not as an essential tool and it can be linked easily dynamically but i.e. a templating engine like Smarty or even more intensive a php-gettext can not easily be "cut out" and the application would probably no more work in a meaningful way! It would also not help if you copy paste GNU-GPL code info an empty file and simply delete the header and no more mentioning the original developer of the GNU GPL code.
i.e. (sorry it is a php-gettext example but which illustrates the main problem of using GPL or LGPL licensed parts in your software product.
Header of the GNU GPL version which clearly states its developers and the GNU GPL license
