It is your tourism that is being applied to that civ. You do not generate tourism for yourself -- think of tourism as the way you exert cultural influence over other civs, while culture is your defense against tourism from other civs.
For example, if you have generated cumulative 10,000 culture, while Civ A has accumulated 1,200 tourism against you and Civ B has accumulated 3,500 tourism against you, Civ A will be exotic to you (their tourism vs. you exceeds 10% of your cumulative culture), but Civ B will be familiar (their tourism exceeds 30% of your accumulated culture).
On the other hand, if Civ C's cumulative culture is only 6,000 culture, and your tourism vs. them is 1.200, you are exotic to them (20% tourism to culture), but if you have a Great Musician perform a concert tour on them that generates, say, 2,000 tourism in one go, your tourism vs. them will zoom to 3,200 and you will instantly be "popular" with them (since 3,200 > 50% of 6,000).
As the article describes, different levels of influence provide different bonuses (e.g., more science from trade routes), but also determine (once ideologies kick in) whether you or the other civ will start to experience dissent, or revolutionary wave, etc., with implications for happiness and whether you (or they) can retain your (or their) chosen ideology.