Answer:
A short way to say Newton's third law is that for every action, there's an equal but opposite reaction. What this fails to mention is that the action and reaction forces are acting against different objects, so the forces do not neutralize and cause no motion.
When you write, you push the pen on the paper; the pen is pushing the paper. Meanwhile, the paper is pushing back on the pen in equal magnitude. The forces balance making the paper stay in place. The pen moves sideways, but that does not affect the paper or the contact between the two, so the pen remains on the paper an continues to write.