How do I quantitatively visualize this situation? Imagine a case where an average of 10% of the income of the people in a certain country is spent on petrol(gas). Now someone comes up with an invention, say a DIY unlimited power source which anyone can make for practically nothing. Obviously the price of oil will fall. But apart from that, how are the other economic indexes going to be affected? Will GDP rise or fall. What kind of parameters of the country will be affected?
Two things will happen:

Firstly, everyone will effectively have had their real incomes rise by 10% (assuming this new energy costs nothing). They will then spend or save this income on other goods besides energy, increasing overall output (note, savings is used for investment as well).

Secondly, energy consumption will likely increase by a large amount now that it is nothing. People who were mindful about their energy consumption due to cost concerns will now not care. Therefore, people will be consuming more energy.