Class Rank is one of the most important forms of progression a character undergoes in MA:O. Class rank determines the crafts you can make, the equipment you can equip, and sometimes the quests you can take on.
To advance your class rank, you need to first level your character to Level 5, as otherwise the class Guild Halls will not accept you. After reaching level 5, you need to find your Guild Hall.
To advance your Class Rank, you need to complete all but one of the tasks available to you at your current rank, at which point you will be unlock a Special Assignment. By completing your class' Special Assignment, you will advance your Class Rank.
As an example, at CR0 your class will have 3 tasks available to them. If you complete 2 of those 3 tasks, you will be given a Special Assignment. You will then need to complete the given assignment, after which your CR will advance by 1.
Special Assignments are unique quests that advance your class' story. Each class' storylines are separate and unique to that class, and can only be experienced by leveling up a character of that class' Class Rank.
The primary advantage of advancing your Class Rank is to wield better equipment. As it is otherwise quite difficult to raise your Equipment Stats, acquiring higher-tiered equipment is the fastest way to increase your damage resistance and damage output.
Beyond that, advancing your Class Rank will unlock new Recipe Challenges, which, once completed, will allow you to craft new items.