My Apple account appears to have a monthly fee of one dollar. How can I understand this or dig deeper?
Apple doesn't charge any fees for using the iTunes or app stores. If you're getting a recurring bill, you probably made an auto-renewing purchase (either a newspaper/magazine app, or some iTunes video content—regular apps aren't permitted to use auto-renewing in-app purchases).

You can manage or turn off these subscriptions in iTunes on iOS or OS X, have a look at Apple's instructions.