The speaker uses alliteration to add emphasis to certain words. Take the first two lines:
Traveling through the dark I found a deer
dead on the edge of the Wilson River road.
The heavy /d/ sound spotlights the words "dark," "deer," and "dead," drawing attention to the scene that sets the poem in motion. Later, the alliteration of the /r/ sound in "River road" creates an interesting sonic connection between two things that are treated as totally incompatible in the poem: nature (represented by "River") and technology (represented by "road"). This sound then gets picked up by "roll" in the next line, subtly suggesting that the clash between nature and technology results in death.
Another prominent instance of alliteration appears in line 4, when the speaker leans on the /m/ sound:
that road is narrow; to swerve might make more dead.
This once again intensifies the speaker's language, all those /m/ sounds calling attention to the idea that leaving the doe on the road could lead to many more deaths.
Other instances of alliteration in the poem include the repetition of the /l/ and /s/ sounds in lines 13 through 15:
[...] lowered parking lights;
under the hood purred the steady engine.
I stood in the glare of the warm exhaust turning read;
The /l/ sound accentuates the words "lowered" and "lights," inviting readers to picture the glow of the car's headlights cutting through dark surroundings. The sibilant /s/, on the other hand, gives this section a subtle hissing sound that evokes the (slightly sinister) hush of the dark night.