Why having two resistors (1k and 10k) in my circuit works?
electricity is resisted to 1K throughout the whole board

Not the whole board - just all the way through a particular path. Both water and the chain belt work as metaphors, but we're not used to chains splitting and recombining.

Recall that electricity flows in a complete circuit: a loop through the power source.

There are multiple complete circuits that current can take through the board, for example:



That path goes through the LED, resistor, and transistor. How much current flows depends on how "on" the transistor is. The resistor is there to limit current when the transistor is "fully on".

Compare:



The blue path goes through the variable resistor, which has a 10k resistor in series with it so behaving like a variable resistor from 10k to 260k. It also goes through the base of the transistor, tending to turn it on.

In both cases the resistors limit the current - to different values of current.

The actual operation depends on variable current flow in the capacitors as the transistors turn on and off; the easiest way to do this is to put it into a simulator like Falstad and watch the result. Astable multivibrator in Falstad. That example uses different values and so oscillates at a different frequency. Try varying them.