As of WordPress 3.5, you no longer need to enable XML-RPC access.
The WordPress app requires a service called XML-RPC on your server. Problems connecting to the XMLRPC service are commonly caused when a web hosting provider blocks XML-RPC. Please check with your hosting provider to find out if they allow XML-RPC on your site.
It can also happen if there is a plugin/theme that conflicts with it. To check whether a plugin is causing the conflict with the app, try deactivating all the plugins on your self-hosted site to see whether that fixes the issue.