Mishna Berura writes (51:1) that it is proper to stand during baruch she’omar. No reason is given. He also writes there, though, that there is a tradition that this beracha was found on a tablet that fell from Heaven. It is probable that to highlight the Divine source of this beracha, we stand as a sign of respect.
Kedushas Yom Tov, a main theme in this week’s sedra, is described as devolving upon the proscribed day “by man” (through kiddush hachodesh), as opposed to Shabbos’ holiness which is “biyeday Shamayim” ; yet they are equated (Rashi Vayikra 23:3).
This should serve as a lesson regarding the kedusha we create in our Botei Knesses and Botei Medrash.