What is the purpose of the area I paint in red in the circuit?

 

Almost right. You put a resistor at the bottom of the LED and amplify and feed back to the 555.
The 2N7000 should get power from the +15V not the +12V.
The Gate drive from NE555 to 2N7000 is not good. You need a P-MOSFET or a new way of driving the Gate above the supply.
Do not need the 5.1V Zener.
There are "LED drivers" that are "buck mode" and "constant current" which would replace the second NE555 + LM311 + 2N7000 + more.

 

Here is one I picked at random. Have used it. LM3404 works from 6 to 42 volts. Works with 1A LEDs. (or less) MOSFET inside.

 

Below is a simple hysteretic PWM constant current supply using an LM339/393, showing powering an LED, discussed here:

 

And your LED is reversed.

 

Almost right. You put a resistor at the bottom of the LED and amplify and feed back to the 555.
The 2N7000 should get power from the +15V not the +12V.
The Gate drive from NE555 to 2N7000 is not good. You need a P-MOSFET or a new way of driving the Gate above the supply.
Do not need the 5.1V Zener.
There are "LED drivers" that are "buck mode" and "constant current" which would replace the second NE555 + LM311 + 2N7000 + more.

The 2N7000 taking power from the + 5V.
I want to set this circuit according to output 1A. Are there any Mosfet models you can recommend for this? and when I install this circuit, should I change the coil and capacitor value?

 

Below is a simple hysteretic PWM constant current supply using an LM339/393, showing powering an LED, discussed here:


View attachment 207543

Thank you for your answer.I will review this circuit.

 

The 2N7000 taking power from the + 5V.

"5V" that is worse than I thought. With the Gate at 5V the MOSFET can only turn on to where there is about 2V on the inductor. The MOSFET will not turn on good.