To course is to run. The most familiar use of this meaning of the word is in “racecourse”: a place where races are run. When the blood runs strongly through your veins, it courses through them. Metaphorically we speak of strong emotions like fear, exhilaration, and passion as coursing through our veins.
Some people mistakenly substitute “curse” and think these feelings are cursing through their veins. This might make some sort of sense with negative emotions, but note that the expression is also used of positive ones. Stick with coursing.