This might help:
For the button "Pay with Bitcoin" to be working you must have a Bitcoin wallet installed on your computer and associated with bitcoin: links. When you click the button, it returns a bitcoin: url, and if no application on your computer is associated with bitcoin: links, then nothing will happen. It is similar like when you click on http: url, it is opened in a program associated with http: links (for example Mozilla Firefox or Google Chrome), and if you don't have a web browser installed in your computer and associated with http: links, then nothing will happen.
If you have a Bitcoin wallet on your mobile device, you can pay by scanning the QR code.
You can see a list of most popular Bitcoin wallets here: https://bitcoin.org/en/choose-your-wallet (select "Desktop" for Bitcoin wallets for desktop) and choose a Bitcoin wallet you would like to use (for beginners I recommend Electrum or Copay).