How can output from a single logic gate/DIP switch supply input for multiple gates?
Remember that voltage is like pressure in a pipe. If the flow from two different pipes is combined, there is more flow, not more pressure. 

Same thing applies when connecting two 5V supplies or signals together, there is more current when paralleling. 

If you want more voltage, you need a charge pump or DC to DC converter.

An Add gate compares the voltages, if both are high at 5V, it wouldn't be possible in the circuit world with switches to get 10V. So instead the additional information is carried as another signal line called the carry out line: 



Source: https://www.electronics-tutorials.ws/combination/comb_7.html