Use it to express moral or emotional backing for someone, especially in tough situations. When everyone doubted her story, I took her part.
Pro Tip 2/3
Not Always Agreeing
Taking someone's part doesn't mean you agree with them on everything, just that you support them in that context. I don't always agree with him, but I took his part at the meeting.
Pro Tip 3/3
Personal Loyalty
It often implies a personal loyalty that goes beyond mere agreement or facts. She's my friend, so I took her part in the argument.