Your site is attempting to communicate with PayPal using an SSL connection (as required by PayPal, since you're sharing sensitive data with them), but your server cannot use SSL over CURL unless your webhost recompiles it to support SSL.
The specific message, according to your logs, is:
You will need to resolve that problem on your host's server, or select another hosting company, before you can use this payment option.(1) libcurl was built with SSL disabled, https: not supported!
Bookmarks