Voltage does not "pass through" resistors, or anything. Nor is it "given off" by capacitors.
Voltage, more rigorously called "electric potential difference", is a difference between two things. When the voltage of only one thing is given, it's implicitly assumed the reference point is an arbitrarily selected node in the circuit we call "ground".
Height is another kind of potential in a gravitational system. Let's say I measure the difference of height between two points on Earth, and I call this the "feetage". The feetage of Mt. Everest is about 29,000 feet, relative to sea level. Does that feetage "pass through" the mountain? Or is it "given off" by the mountain? No, that doesn't make any sense.
Pressure is another kind of potential. For pressure we can define a vacuum as zero pressure, so we can talk about absolute pressure. We can also talk about the difference in pressure between two things. For example the difference in pressure between the air in your lungs and the air around you is usually very small, because when it isn't air moves in or out of your lungs via your nose until the pressures are equal. Does the pressure "pass through" your lungs? No, it's the air that passes through.
The "air" in an electrical system is called electric charge, measured in coulombs. The rate at which electric charge is moving is called current, measured in amperes.
If the passage of air is restricted, then the pressure difference will be greater, and less air will flow. If the passage is nearly unrestricted, then the difference in pressure will be very little, and much air will flow. This is resistance, and it works similarly in pneumatic, hydraulic, and electric systems. Resistance doesn't really regulate current: it just determines at what rate fluid (air, hydraulic fluid, electric charge, etc) will flow for a given difference in pressure or electric potential (voltage) between two things. This relationship between voltage, current, and resistance is given by Ohm's law.
What the 555 timer is looking at is the difference in electric potential between the bottom of C1, and the top. The voltage across the capacitor is the integral of the charge that's passed through it: passing one way the charge increases the voltage, and passing the other way decreases the voltage. And in turn the charge passing through C1 is a function of the battery which is pumping charge through R1 in one direction, and the 555 which is letting charge flow out in the other direction (or not, depending on which state it's in).
Hopefully that makes things more clear.