All birchos hamitzvos must be said while standing. The word v’tzivonu (“and He commanded us”) connotes that we are in Hashem’s service; just as the avodah in the Beis Hamikdosh was done while standing, we stand when calling attention to our own service. In addition, sitting during the brochah gives the impression of denigrating the mitzvah. Those who are fulfilling their obligation by listening to the brochah must also stand during its recital.
Although, as explained, a person is allowed to sit during the tekiyos d’meyushav, it has become customary for the tzibur to stand while listening to the blowing. Some Achronim explain the reason as follows: A person must not talk from when the brochah is recited (before Musaf) until after the tekiyos d’meumad are blown. Since it is difficult for people not to talk, the primary tekiyos were switched from being blown during Musaf to being blown before Musaf. As such, the tzibur must stand while they are blown.
[משנ"ב תקפה, א; ביאורים ומוספים דרשו, 1, 3 ו־4, בהרחבה ע"פ המקור]