In order to download and install Dreadnought for the PS4, visit the store on your PS4's Home page as shown below.
Next, type in "Dreadnought" in the search bar and select the game.
Once you have selected "Dreadnought" click the "Download" button on the content screen.
You can check the progress of your downloads in your Library or main Game Menu.
Complete the installation progress and log in to begin blowing stuff up! Welcome aboard, Captain!