"How come" vs "Why?"

What are the differences between the terms "How come ... we eat breakfast?" and "Why ... do we eat breakfast?"

The words phrase based in how seems really awkward to me, and I don't understand this convention.

"How come...we eat breakfast?" is less formal. As Robusto commented, it's a contraction of something like: "How does it come to be that ...?" or "How has it come to pass that...?"

Also, it's not as confrontational. "How come you turned up late?" is softer than "Why did you turn up late?". It is a subtle mechanism that allows us to drop the 'do' verb from the question.

The latter is something your boss might ask. It is a very direct question expecting a direct answer, with the emphasis on "you doing".

The former is something your work colleague might ask. It is softer because it acknowledges that 'it came to be' that you were late. The emphasis is shifted off "you doing".

There are answers here that are close, but claim that "how come" is a contraction.

Come has a sense, meaning "turn out", "happen", "come about". It's a relatively rare use now, but it was once more common:

Til it com on a fest dai, þat king herod did for to call þe barnage — Cursor Mundi c 1400

Whan it came vpon a daye that Elcana offred. — 1 Samuel 1:4, Coverdale's Translation, 1535.

All things ar cumde for the best. — Ane verie excellent and delectabill treatise intitulit Philotus, 1603

And when that sense was current, we could apply how to it to enquire as to the way by which something "turned out/happened/came about":

How com'st that you haue holpe To make this rescue? — Shakespeare, Coriolanus.

While this sense of come waned, its use in how come popped up as a colloquial survivor and grew from there.

Considering it a contraction of "how did it come that" will provide a fine understanding of the meaning, but is not what it actually is.