How the transistor works in this 555 circuit?
When the button is pressed the capacitor is charged instantaneously to 8.74 V. It will then turn on the transistor and a current will flow through RT, the value of this current is (VC-0.6)/RT where VC is the voltage in the capacitor. 

When the button is released the transistor will be kept ON by the capacitor and a small fraction of the previous current \$\approx \frac 1 \beta\$ is provided by the capacitor itself, slowly discharging it. 

The final effect is that the capacitor is discharging through an equivalent \$ \beta \cdot RT\$ resistor instead of the \$RT\$ resistor you would have without the transistor. Because \$\beta\$ in transistors is around 50-150, you can have very long discharge times.

The leakage current of the pins in 555 (trigger and threshold) puts a limit on the maximum effective resistor you can use as RT (because the leakage can be thought as a parallel resistor internal to the 555). Your configurations allows to multiply this value by \$\beta\$.