What are Tareme and Tsurime eyes?
You can read about them in detail on TVTropes:
A particular style of drawing eyes in manga and anime where the outer corners slant upwards, ending in a distinct sharp point, used in contrast to the traditional Moe soft drooping style, Tareme. Tsurime symbolizes strong will, arrogance, or pride in a character. Most noticeable in female designs (since it plays against the stereotype) and is commonly an attribute of Tsunderes, Kuuderes and particularly Badass Action Girls.
A style in anime and manga of drawing eyes where they droop or sag at the corners. Typically, a Moe character will have those definitive, sympathy-magnet Puppy-Dog Eyes. It generally represents a kind, quiet, sad, fragile or otherwise soft person. Only really notable if there are more than a handful of characters that are drawn without them.
As you can make out, Tsurime eyes are the opposite of Tareme eyes. As for "why" the characters have it, it's a symbolization used to depict their characteristics which is easily understood by a regular anime watcher.