What made these burrows and holes in my lawn (Picture inside)
Those look like mole runs under the sod. That is how they move around. They eat grubs and worms just under the root layer of the grass. Eliminate their food source and they will leave. 

You can treat the lawn with grub killer or nematodes that kill the grubs. I have had these for years. They come and go but when I treat the lawn they seem to leave for a season.

Good luck!