Why do series resistors have a single current running through them?
Suppose some charge exits the right terminal of R1, where can it go besides into the left terminal of R2?

If the current exiting the right terminal of R1 isn't equal to the current entering the left terminal of R2, then charge must be accumulating in the wire between the two resistors. 

To model that possibility, we could include a parasitic capacitor between that node and the ground node in our model. Then we'd see that when that happens (assuming the current through R1 is greater than the current through R2), the voltage on that node will increase. Which will cause the current through R2 to increase until things balance out again.

By not including the parasitic capacitor in our  model, we're saying that this capacitance is so small as not to matter for the way we're using the circuit. And any capacitive charging needed to get the resistor currents balance happens so fast after connecting the 9 V source that we can ignore it.