How does a jet's throttle actually work? What does it do to the pressure chamber and the fire output controller?
The throttle (or thrust lever) in a gas turbine engine adjusts the thrust produced by controlling the fuel flow to the combustion chamber.
Most of the modern engines are actually controlled by the Full Authority Digital Engine (or electronics) Control, FADEC (or Engine Control unit)- so the signals from the throttle are sent to the computer, which regulates the fuel flow based on various parameters (including engine safety) thereby adjusting thrust. The computer handles the additional components of the engines like the thrust reversers and afterburners.
The image below shows the throttle controlling the fuel system through the engine control in the Pratt & Whitney F100 engine that powers F-15 and F-16.
Image from kaskus.co.id
I'd suggest that you review the topic of how a jet engine works, first, and then review how a jet engine fuel control unit works as it doesn't sound like you're very familiar with engine components; I've never heard of a 'pressure chamber' or 'fire output controller'.
Modern jet aircraft technically don't use throttles; rather they are equipped with thrust levers which are connected to a Fuel Control Unit (FCU). This can come in the form of a mechanical computer in the case of earlier engines or a digital electronic computer called a Fully Authority Digital Engine Controller (FADEC), similar to the schematic in aeroalias's answer for the F100 fighter engine installation on the F-16 airplane.
Early jet aircraft did use a throttle - literally a cockpit lever connected to a throttleable valve to precisely meter fuel into the combustion chamber(s) of the engine. This could be problematic to operate in flight as the fuel rate has to be changed as altitude - and consequently combustor inlet conditions - changes, making the engine vulnerable to flameouts from excessively rich fuel flow and compressor stalls from abrupt throttle changes. FCUs were developed, specific to each engine to offer simplified control.
Jet engines run on the Brayton cycle which is isobaric (constant pressure) combustion from a thermodynamic perspective.
Assuming the engine is already in a steady state, when you open the throttle you introduce more fuel (regardless of how computerised or otherwise).
More fuel translates as more heat into the combustion chamber. As we calculate this as constant pressure this mean we increase the volume of the fluid flowing through. In practice this is seen as a faster fluid flow rate exiting the chambers This increased volume/flow rate impinges on the turbine as it expands adiabatically it imparts more work into the turbine, spinning it quicker which spins the compressor quicker drawing in more airflow. Once past the turbine the exiting gas flow has a higher velocity and higher momentum which through Newtons laws becomes greater thrust. It's not finished there through. The turbine/compressor will continue to speed up (adding more air to the system) until the system achieves a new steady state at a higher fuel flow, higher compressor/turbine rotation speed, higher turbine temperature and of course higher thrust. We could keep adding higher and higher fuel flows but there are a number of limits to the system, which ultimately limits the maximum thrust.