I think the OP is asking what should happen with the LEDs as the temperature changes.
With reference to the original circuit, the output of the LM35 rises as the temperature rises.
TEMPERATURE BELOW THRESHOLD
Assuming that the LM35 output (connected to pin 3 of the comparator) is lower than the voltage you have set on pin 2, then the voltage on the comparator output (pin 6) will be close to ground. In fact it needs to be less than 0.6V so that it doesn't send current into the base of Q1. In this circumstance:
1/ Current flows from the 5V rail through R4, D1, R5 and into the base of Q2, switching it on, so that current can flow freely from Q2's collector to its emitter. Thus current can flow from the 5V rail through R6, D2 and Q2 to ground, lighting up the green LED (D2).
2/ The base current that switches on Q2 flows through the red LED, which is likely to glow dimly as modern red LEDs are very efficient.
TEMPERATURE ABOVE THRESHOLD
When the temperature rises such that the input voltage to pin 3 of IC3 goes higher than the reference voltage on pin 2, the voltage on IC3's output pin (pin 6) switches high - that is, switches to a voltage just a little under 5V.
This forces current into the base of Q1, switching Q1 on such that current will flow freely between its collector and emitter. This time, the current through R4 and D1 has an easier path straight to ground (via Q2) compared with having to go through R5 and Q2. Thus two things happen:
1/ Current flows freely through R4, D1 and Q1, lighting up the red LED (D1) brightly.
2/ No current flows into the base of Q2 (because it's all gone through Q1 to ground), switching Q2 off, and therefore no current can flow through R6 and the green LED. Thus the red LED glows brightly and the green LED is off.
As you have already read, there are problems with this circuit, not least that the red LED will probably glow dimly even when it should be off. It also requires a comparator that will drive its output close to the -ve rail in order that Q1 remains completely off under low temperature conditions.
Anyway, that is how the original circuit should behave, which is what I think you were asking in your first post.