Why were so many European airports closed due to the volcano? The image above of one volcanic ash particle begins to tell us why: the extremely small particles, with their many voids, can travel great distances after eruption. Once inside a jet engine, they melt and then re-solidify. Read Properties of Volcanic Ash for more details. You can learn about the specific dangers of flying through volcanic ash here.
Airlines try to avoid flying through ash, but it does happen. Read the account of one pilot at What it's Like to Fly through Volcanic Ash, and see the precautions against ash that Boeing takes here. For more on the hazards of flying through ash, see this Cocktail Party Physics blog post.