So, I'm a third year student studying mathematics. My goal has always been getting a PhD in math. I'm currently doing ok with my math (taking grad courses and stuff). However, as many other students, I have to take courses outside math. I'm currently taking a labor studies class right now. All the class talks about is how bad America is treating its workers, and why don't we raise minimum wage to 50 dollars per hours to help the poor workers and stuff like that. I don't agree with everything I learn in class, and I show my disagree. As a result, the professor gives me terrible grades for saying my opinions. My question is: Will that bad grad hurt my future? Can I explain my situation in my application?
This could possibly hurt you.
In grad school, and even more so as a working academic, you will be asked to do various tasks that you might disagree with. For example you might be asked to teach precalculus or college algebra. Occasionally grad students don't like how they are asked to teach these courses and they either raise a big stink, or else simply don't follow instructions. In either case, this can cause huge headaches for the person supervising the TA -- even if the TA's complaints have some merit. Whether fairly or not, the prospect that you may be unwilling to follow instructions might give some admissions committees pause.
Doing tasks which you disagree with and that were assigned by people whom you may or may not respect (and doing them quickly, so that you can get on to the work you actually care about) is a vital survival skill in academia. I recommend that you learn this skill immediately and do what you can to pass the course.