Then he heard the slithering. A brushing sound, a slithering brushing sound near his feet—and he kicked out as hard as he could, kicked out and threw the hatchet at the sound, a noise coming from his throat. But the hatchet missed, sailed into the wall where it hit the rocks with a shower of sparks, and his leg was instantly torn with pain, as if a hundred needles had been driven into it.
"Unnnngh!"

What does "out" mean or imply as in "kicked out"?

Question

Then he heard the slithering. A brushing sound, a slithering brushing sound near his feet—and he kicked out as hard as he could, kicked out and threw the hatchet at the sound, a noise coming from his throat. But the hatchet missed, sailed into the wall where it hit the rocks with a shower of sparks, and his leg was instantly torn with pain, as if a hundred needles had been driven into it.
"Unnnngh!"

What does "out" mean or imply as in "kicked out"?

Answer

Kick out - to move something (which is near you) away with force using your legs (literally or as a figure of speech).

It can also mean to forcefully refuse.
 Eg The senators kicked out against the student debt forgiveness bill.

Answer

In the case of your paragraph ‘kick out’ means to kick your leg out in front of you.  It can also mean to make someone leave ie.  I kicked him out of my apartment