Here is a version of the circuit that simulates ok, and should work when breadboarded. I fixed some things I didn't like about the circuit you downloaded ...
Here are my assumptions:
1. The resistance of the LDR when the circuit switches is 100K. If it isn't, then adjust (my) R2 to match whatever resistance you wish to define as the transition from dark to light...
2. U2 is a comparitor whose output pin pulls below 0.3V when low. If it doesn't have a built-in pull-up, you need to add (my) R6.
Here are the things I fixed:
1. There was no hysteresis around the comparitor. I added R5, which provides about 45mV of hysteresis at the comparitor's input
m. That prevents the comparitor from "chattering" due to 120Hz flicker in any illumination falling on the LDR. If it still chatters, reduce the value of R5...
2. I juggled the timing network (my) R7, R8 and C1 to values that will make the battery last longer.
3. I added the missing power supply bypass C3, and the missing bypass C2.
Attached is my LTSpice file. Note how I modeled the varying LDR resistance by making it a function of a ramped voltage which has some 120Hz ripple injected into it. Simulate it with and without R5 to see how converting the comparitor into a Schmit Trigger with 45mV of hysteresis gets rid of the multiple switching you were likely experiencing with your circuit...