Get the latest release of the Ratepay payment extension for Magento 1 from https://github.com/ratepay/magento1.7-module/releases and store it locally.
Enter the Magento Connect Manager. Go to the menu System -> Magento Connect -> Magento Connect Manager.
Login with the admin-credentials.
In the Magento Connect Manager in the section Direct package file upload click on Search... (DE: durchsuchen...) and select the TGZ-File that contains the Ratepay Payment Extension. Click the button upload to start the installation.
The installation procedure may take some time. After a reload in the bottom of the page shall be displayed:
After clicking the button Refresh there shall be a new line at the very bottom of the table starting with RatePAY_Ratepaypayment in the first column:
The version-number in the second column may vary as you may use a newer release of the extension.
To check the correct installation click on the admin section and have a look at the menu for an Ratepay entry.
Go in the menu to System -> Configuration -> Payment Methods. On the page loaded there shall be shown a new section for Ratepay: