You need to do this, as explained in the itch.io semi-portable instructions:
Install the following libraries (execute in terminal):
sudo apt-get update
sudo apt-get install libsdl2-2.0-0 libopenal1 libcurl3
Then, say you extracted the files to
/home/user/zero-k-portable
folder, you do
cd /home/user/zero-k-portable
./Zero-K_linux64.sh
Later you will probably want to create icon/shortcut for this, but first see if it works the basic way.
P.S. why Mint?