In your ~/.zshrc
Simply clone this repository and source the script:

git clone https://github.com/zsh-users/zsh-syntax-highlighting.git
echo "source ${(q-)PWD}/zsh-syntax-highlighting/zsh-syntax-highlighting.zsh" >> ${ZDOTDIR:-$HOME}/.zshrc

Then, enable syntax highlighting in the current interactive shell:

source ./zsh-syntax-highlighting/zsh-syntax-highlighting.zsh
If git is not installed, download and extract a snapshot of the latest development tree from:

https://github.com/zsh-users/zsh-syntax-highlighting/archive/master.tar.gz
Note the source command must be at the end of ~/.zshrc.