The Greeks have always been fascinated with the human form, but it isn't until the High Classical period that one begins to see true idealism. There is a fascination with the musculature of man and how he looks in motion. There is a particular interest also in the male form depicted at his most beautiful. In fact, the musculature of the Boxer is impossible; he is flexing opposing muscles while at rest, something a human can't do but sculptors took liberty with in order to demonstrate the idealism in the power of the form. 

The work depicts not a god but a man, and therefore would not have been in a temple. His form is athletic and he is nude, as male athletes would likely have been, but would not have been appropriate for an open public space. He's made of bronze and therefore likely too expensive for any patron other than the state. This leaves the gymnasium as a likely home, especially given it was a place of learning and fitness where only men could go. 

The Greeks, especially Athenian Greeks, were accepting of practicing male homosexuals as it was often seen as a venture of the stronger sexes with no interference from the women. Athletics especially were a medium of this expression. This translates into Greek art wherein male athletes depict the same homoerotic undertones that would have colored the actual event in much the same way.

High classical sculptors in the Greek tradition were more interested in the male form and its depiction in motion than expressing a particular emotional quality. Statues of the classical tradition were known for their muscular frames but a disinterest in the face. The expression was largely blank, meant to direct focus on the body instead.

Most figures in the Greek artistic tradition were nude, especially the men. It was meant not only to demonstrate the form in motion (best seen in the nude) but also as a mark of tradition. Athletes competed nude in Greece, even boxers. The sculptor wouldn't have had a reference as to what he would wear, anyway. 

The archaic tradition in Greece was the first time Greeks expressed interest in the human form and its musculature; however, those statues originated not in Greece but in Egypt, where the Ka statues represented the dead and were the first recorded instances of upright human figures that were almost anatomically correct.

First take note of the form and composition of the piece. The boxer has correct proportions- he is idealized and athletic. All of the answers allude to this. Take special note, though, of his emotional appearance. He is not stoic, and certainly not sultry. And since the use of bronze is not novel for Hellenistic Greece, the answer must be that he is in anguish, though perfectly moulded.

The boxer is not carved from stone; he is not carved at all. The sculptor used the lost wax process wherein the piece is made from bronze. Note the green tinge, indication of the oxidation of the metal. This is standard for all Greek pieces (it was the Romans who coped classical works using marble instead of metal).

The development of deep drillwork, which made the "corkscrew" curls of this portrait possible, occurred during the Flavian dynasty (Roman Early Imperial.)

Image: Portrait bust of a young Flavian woman, potentially Julia, daughter of Titus. Marble. c. 80-90 BCE. See page for author [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Matronalivia2.jpg.

Contrapposto was the signature pose used in Greek and, through Grecian influence, Roman statuary. Literally the Italian for “counter-pose,” The sculpture of The Augustus of Prima Porta follows contrapposto perfectly. Augustus has one foot in front of the other, with his shoulders turned to face the viewer, creating tension and movement in the overall pose of the figure but a relaxed posture in the subject.