Q:

Why did standing on one's feet become the posture for prayer?

When one reads about people praying to G-d throughout Tanac"h, we rarely discover them standing to pray. Most of the time they are kneeling, prostrating or falling on their faces. We see that when Moshe prayed, he fell on his face (See Bemidbar 14:5, Bemidbar 16:4 for two of many examples.) 
In Daniel 6:11, we see that he faced Jerusalem and prayed 3 times daily in a kneeling position. The 3 times daily prayer facing Jerusalem looks like the closest "imitation" to our current Amidah (in terms of facing Jerusalem.)
There are some exceptions. Regarding Avraham it says, "He was still standing before G-d.' (Breishit 18:22)
The Gemarah pretty consistently says that prayer (generally, in the Gemarah, the term tefilla usually means the Amidah aka "Shemoneh Eseh") should be done standing. See Mishnah Bavli Brachot 3:5 and Brachot 5:1 which both use the term "standing".
At what point and why did the posture change to standing?

A:

Ma'aseh Rokeach to Hilchot Tefillah 5:2 writes that a source for the requirement to stand during prayer is the concept of:

אין שירות אלא מעומד
The [priestly Temple] service was only performed while standing

(mentioned in Sotah 38a).
My understanding of his comment is that our prayers today are modelled (to an extent) on the Temple service. We therefore stand to pray, as the priests stood to serve in the Temple.

