The kohein (priest) would slaughter the bird by severing the trachea and the esophagus with an overgrown thumbnail in a process called melikah. He was not, however, to cut all the way through and cause the head to be completely separated from the body.
The Sefer HaChinuch says that we can never hope to completely understand the reasons behind all the details of the sacrifices, this one among them. However, so as not to leave us hanging, he gives us a lesson from this mitzvah. As described in the previous mitzvah, the birds were brought as an offering by one who didn’t have the means to bring a sheep or a goat. From the fact that the bird is slaughtered with a thumbnail rather than a knife, we see that one must act with zeal to attend to those in need. Rather than making the poorer person wait until he got a knife, the kohein would slaughter the bird with what he literally had at hand: his thumbnail. (He also slaughtered the birds from the back of the neck, which was faster. Plus, the bird could be sacrificed at any place around the altar; the kohein did not have to bring it to a specific corner. This, too, sped up the process.)
Another lesson the Chinuch derives is that the Jewish people are compared to a dove (see, for example, Psalms 68:14). Severing the necks of the sacrificial birds is a metaphor not to be stiff-necked.
But why not separate the head? In order to keep the sacrifice in presentable condition. A single bird is better than a bird body in one hand a bird head in the other. It’s bad enough that this person had to bring a second-choice sacrifice; the least the kohein can do is keep it in good condition in order to make it a nicer sacrifice.
This prohibition applied in Temple times and not only for kohanim; anyone who severed the head of a sin-offering bird would be liable. It is discussed in the Talmudic tractate of Zevachim on pages 65b-66a and elsewhere. It is codified in the Mishneh Torah in the seventh chapter of Hilchos Maaseh HaKarbanos. It is #112 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.