Based on the Bavli (B'rachos 34), Shulchan Aruch (OC 113) rules that we must bow at the start and end of the first and penultimate b'rachos of sh'mone esre (or amida). This is the general practice, too. Why those four points in sh'mone esre?
The Maharal in Nesiv HaAvoda ch. 10 writes (copying bits and pieces because it is lengthy):
To summarize (with some additions I didn't copy, it's a long piece):
The first blessing includes the "force" of the entire Tefillah, because the beginning includes everything that follows it. Therefore a person is exceedingly close to Hashem in this blessing, and you bow to the King when you are in front of Him. We are acknowledging that Hashem is the source of everything, the G-d over the beginning itself. By bowing, we acknowledge that Hashem is the only existence and we nullify ourselves to Him.
So to at the end [of the first blessing], because the Avos are the beginning, as has been explained.
Same with Modim, we acknowledge that all existence is Hashem's (as we say, our lives which are given into your hands etc.).
As to why it is at the beginning and end of those blessings, that is because:
When we come to mention of Hashem's name, we bow before it to show that our existence is nothing in context of Hashem's existence. When we reach Hashem, we straighten up because His existence gives us reality.
The idea of our bowing is to show that we are nullified to Hashem and His existence is the only real existence, as above. Hashem's name is at the beginning and end of each of those two blessings, so we bow there.
I recommend seeing the whole piece inside to understand how everything connects (plus really appetizing extensions of the idea).
so it seems it is a matter of knia (submission/humility towards God)
they are in the beginning and end of the amida, as above. beginning is humility and likewise at end when you part, just like when you leave the western wall.
besides the above quote, it is logical since when you speak to a great king you must show humility before and after, how much more so to God.