Here's my suggestion.
The 555 produces a rectangular output wave with a period of about one second. Each time the output changes state, C3 charges up or discharges, and there is a burst of voltage across R3 as it does so. The four diodes form a bridge rectifier that converts that voltage pulse from bipolar to unipolar so the LED(s) will light on each transition.
R4 limits the LED current. The LED (with R4) is almost in parallel with R3, and the LED current has a bigger effect on the pulse duration than the value of R3.
The timing of the blinks is determined by R1, R2 and C1. The short time between the two blinks is proportional to R2 and the time from one pair of blinks to the next is proportional to R1+R2. Both are proportional to C1.
C2 is a decoupling capacitor to help the 555's stability. CDRIVE, I've taken your advice on this one
Reducing R4 will increase the peak LED brightness but also shorten the blink duration, so the apparent brightness may not change much. Increasing C3 will increase the apparent brightness though, by increasing the blink duration without affecting the peak current. With the values given, the time for the LED current to fall from its peak (~100 mA) to half (50 mA) is about 1.4 ms. Watch the peak current specification of your LED; LEDs can be damaged by overcurrent. Also, the 1N914 diodes and the 555 are only rated for 200 mA so limit the peak current to that, or use a buffer and bigger diodes. If you need more apparent brightness, your first change should be to increase C3.
Varying R3 will make little visible difference. Decreasing it significantly will shorten the blinks and reduce the peak LED current. Increasing R3 will slightly lengthen the blinks and slightly increase the peak LED current, but may cause the end of the blink to become less clearly defined and look more like a fade-out than an abrupt cut-off.
You can connect multiple LEDs in series, but if the total forward voltage becomes significant (more than about 3~4V at 12V supply) the current will start to drop noticeably and you will need to reduce R4. I've recommended a 12V DC supply; up to 15 is OK but if you use less than 9V you'll need to reduce R4 and the pulses may become less clearly defined, so I don't recommend using less than 9V.
The part number for the LED shown on the schematic has no significance. LTSpice has a very limited range of LEDs so I just chose one that looked kinda suitable for my simulation.