how amperage of the source influences the components of a circuit?
Why does all this mention the source current? isn't the source supposed to only provide voltage and the current is defined by the resistance of whatever im connecting to the source?

What the power supply is saying is that a load (the device that draws power from the supply) may safely draw up to the rated current and still assume the voltage will remain within the stated specifications.
So a &quot;5V 1A&quot; power supply should maintain an output voltage of 5V if you don't draw more than 1A from it.

But if I connect the same resistor to a usb port that &quot;supplies 5v and 1amp&quot; the circuit current should be 2 amps anyway because V = IR and with that the resistor will dissipate 10W because P = V ^ 2 / R, right?

All bets are off if the load draws more than the current rating of the supply. Usually one of following happens in such cases:

The output voltage decreases.
The power supply may simply shutdown and stop supplying current. The shutdown may be temporary or last until some reset action is performed by the user.

Some computers will even initiate a shutdown of the system if an overcurrent event is detected on a USB port.
Power supplies which don't protect themselves may even get damaged if you try to draw too much current from them.