I have decided to purchase a real time clock module for an automated feeder project I am attempting. I would like it to come on everyday for 7pm using a servo to open a door for the food to come out but I am having some difficulty understanding what is meant by "The DS1307 is accessed via the I2C protocol". In Layman's terms, how exactly do I reference the library to read the time? Do I need to purchase additional hardware?
The DS1307 real time clock chip is a 8 pin IC that uses two wire (plus ground) communications to a host device, such as an arduino, using the I2C communications method which the arduino supports directly. So no you don't need any additional hardware other then the DS1307 module.
What you do need is software support in the form a a library that allows you to utilize the DS1307 in your arduino sketch to utilize it. There are several 3rd party libraries that will provide this support but you have to locate them, download them and learn how to utilize their commands, as the Arduino IDE comes with no standard library to support this RTC module. A DS1307 support library will handle all the details of how to communicate between your arduino sketch and the RTC module using the I2C communications protocol. You just need to study the library you settle on to see how to use it in your sketch.