How is data added to the Health app? I've been very late at updating my phones / OSes. 
Now there's a new app called "health" that I have no idea how to use. I usually understand Apple products and apps very quickly, but I am at loss of ideas on how one adds data to the graphs in the Health app. Can the iPhone itself track your steps / physical activity? Or do I need other hardware such as watches and things like that?
It depends on your iPhone model. The iPhone 5S and later is equipped with a motion co-processor, such as the M7 in the 5S or the M8 in the iPhone 6 and iPhone 6 Plus. You can give Health permission to read data from the motion co-processor.

Also, any data from your Apple Watch (if you own one) is stored in Health. So heart rate, steps taken, walking + running distance, active calories, etc. are counted automatically.

Additionally, 3rd-party apps can use Health to log any appropriate info about your health and activity into the app, such as Nike Running or an app that can detect your pulse using the camera in your iPhone.

You can also log data into Heath and modify data yourself within the app.