It expresses strong feelings. Use it when you have a serious issue with something said or done. She took exception to the way the article misrepresented her views.
Pro Tip 2/3
Subjective Disagreement
It doesn't mean something is objectively wrong, but you personally find it upsetting or incorrect. John took exception to the critique, feeling it was unfair.
Pro Tip 3/3
Politeness
While strong, it's not rude. It's a formal way to express that you are upset or disagree. I must take exception to your comments about my work.