Sure, your Moto X Style has a support for USB OTG. Connect your external hard drives to the device and check the files stored on them on your phone. Some file formats may not be supported on your phone. Please ensure you have backed up your important files before you resort to format your external hard drive to the required file format.
Moto X style does support OTG cable. As per box manual's statement, the OTG support is official. You can expect the future versions of the mobile to have the support for OTG cable too.
You can also view how to use OTG cable with Moto X Style on this video: https://www.youtube.com/watch?v=lj4ccFH1jiw