This theme works only if VIM supports termguicolors option. This is true for Neovim and VIM from 7.4.1799.

Plug 'ayu-theme/ayu-vim' " or other package manager
"...
set termguicolors     " enable true colors support
let ayucolor="light"  " for light version of theme
let ayucolor="mirage" " for mirage version of theme
let ayucolor="dark"   " for dark version of theme
colorscheme ayu