The Spanish hinchar means “to inflate, puff up” and from it we get the much more common Spanish hincha meaning: “a fan”. A fan, therefore, is literally someone who puffs himself all up over his team!
Interestingly, hinchar is directly related to inflate, in a subtle way: both come from the Latin inflare meaning the same as hinchar.
How did this word evolve into somethings so different? It’s not so different as it sounds if we remember that the Spanish h- is silent: so the in-ch‑a maps closely to the in-fl‑a. The ch/fl mapping isn’t common at all, but if we sound it out, we can hear that they sound similar.
Next time you get all excited about your favorite team, remember that it is this excitement of making more and more is exactly what causes… inflation.