And his sister stood from a distance'...why did Miriam stand from a distance? Rabbi Amram said in the name of Rav: Miriam had a prophecy. Sharing it she proclaimed, 'my mother is destined to give birth to a son who will save the Israelites!' When Moshe was born, the house was completely illuminated* signaling a miraculous birth. Her father stood up and kissed her on her head. He said to her, 'my daughter- your prophecy has been fulfilled'. This is the menaing of the verse (Exodus 15, 20): And Miriam the prophet, sister of Aaron (sic!), took the drum'- Aaron's sister and not Moshe's. When she saw her prophetic vision, she was still only the sister of Aaron, Moshe was not yet born. When they threw Moshe into the Nile, her mother stood up, slapped her on her head and said to her, 'my daughter, now where is your prophecy??!' This is the meaning of the verse, 'And his sister stood from a distance, to know what would happen to him'. To know the end of her prophecy.
ותתצב אחותו וכו'. מהראו שיאמר מה יהיה מהנער. אך הנה ארז"ל (סוטה דף י"ב) שטרם יולד התנבאה ותאמר לאביה ולאמה הנה בן תלד אמי שיושיע את ישראל. וכשהושם ביאור טפחוה על ראשה ואמרו לה היכן נביאותיך. והנה אין ספק כי עודנה מתחזקת באמונת נבואתה ע"כ אומר הכתוב כי הנה ידעה בודאי שיעשה לו נס. אלא שלא ידעה מה יהיה הנס. וז"א לדעת מה יעשה לו כלומר מה יהיה הנס אשר יעשה לו:
...and without a doubt that Miriam was still believed completerly in her prophecy, this is why the verse uses the word, 'to know', ie she knew that a miracle would take place but she didn't know how the miracle would transpire.
...'she stood' is a strange and inaccurate grammatical form. It should have written, 'va'tit'yatzev'. The meaning however is that she was 'placed', by an other, i.e. by Divine Providence.
...This entire verse is referring to the Shechina, the soul is a part of the Divine and this aspect of Divinity that watches over each of us is like a twin sister to our own soul that are never separated (Siamese?Conjoined?). Even when she stands at a distance, she watches over and protects.