Milah enables one to bear revelation

ברש"י עד שלא נימול הי' נופל על פניו כשהי' רוה"ק שורה עליו.

In Rashi: before he was circumcised, he would fall on his face when ruach hakodesh (the holy spirit) rested upon him.

Rashi notes that prior to his bris milah, whenever the ruach hakodesh came upon Avraham he was overwhelmed and fell on his face, unable to remain standing before the Divine revelation.

הגם כי מקודם הי' לו ג"כ מראות ולא נפל על פניו רק שאין כל המראות שוות ועתה הי' לו מראה בזה שניתן לו מצות מילה ולא הי' יכול לעמוד עד שנימול.

Even though earlier he had also had visions and did not fall on his face — for not all visions are equal — now he had a vision in which he was given the mitzvah of milah, and he could not remain standing until he was circumcised.

Avraham had received prophetic visions before without being overwhelmed. But this revelation was of a higher order: it carried within it the mitzvah of milah. Such an elevated light could not be borne by his uncircumcised body — only once he was circumcised could he stand before it.

וכן מ"ש בבלעם נופל כו' ע"י שהי' לו אז מראות בעבור בני ישראל:

And likewise what is written regarding Bilaam, "fallen…" — because he then had visions on behalf of Bnei Yisrael.

The same principle explains why Bilaam is described as prophesying while "fallen": the visions granted to him concerned the greatness of Bnei Yisrael, a light too lofty for him to receive while standing, so it forced him down.

Summary: A revelation carries a measure of light according to its content. Before his bris, Avraham could not stand before the elevated vision that contained the mitzvah of milah and fell on his face — only once circumcised could he bear it. The same reason explains why Bilaam prophesied "fallen," since his visions concerned the exalted greatness of Bnei Yisrael.