Maybe these user-friendly translations will help!
Purpose: The purpose of your writing is just what it sounds like--it's the point of the piece. Each writing will have a different purpose; it's up to you to make sure you keep your eyes on the prize. That means remembering who is meant to read it and what it is aiming to achieve. For an academic piece, be sure to maintain focus on the claim, and use language that is appropriately scholarly. Writing a creative piece? What meaning are you trying to convey to your reader? Have you given him or her what they need to engage in your work?
Thinking: Again, this is just what is sounds like! To what extent does your writing reflect your own ideas and critical thinking? Did you just repeat what the teacher said or summarize what you read in the text? Or did you take a risk by making a new claim, attempting a novel approach to storytelling, or presenting your unique perspective in original, convincing language?
Organization: Starting to see a trend? Yes, all writing--from a Chemistry Lab to a response to literature--should be organized in a way that helps convey your meaning. Sometimes that organization resembles typical essay structure, sometimes it is an organization of your own design. The most important thing is that you have considered how you are going to present your ideas.
Language: This category measures the quality of your language. Are you using words that are appropriate for the content? Do you vary your sentence structure to match the information being conveyed? Here is where the sophistication in your use of language is really assessed.
Mechanics: And here's where the rubric considers whether or not you've crossed your t's and dotted your i's. Are your punctuation and grammar correct? Is your spelling impeccable? Do your style choices align with standards?