Logic behind this fuse status indicator
Q1 and Q2 form a astable multivibrator which will output a square wave (12V, 0 V, 12 V, 0 V, ...) voltage via D2. If D1 wasn't there then that square wave would turn on/off Q3 and with that also the LED. So the LED would blink.

However D1 is there and via R5 it keeps Q3 on even if the square wave voltage is 0 Volt. So the LED will not blink, it will be on continously.

But there is more. I didn't discuss fuse F1. If that fuse blows (due to overload or short circuit) then there will no longer be 12 V at R5 so R5 and D1 can no longer "overrule" the square wave voltage when it is 0 V. So that means that the LED will blink.

So:

Normal operation, fuse is intact: LED is continuously on

No power at output because the fuse has blown: the LED will blink

Oops, forgot the 3rd state:

When the LED is off, the battery is empty (assuming the circuit and LED are still working).

Sidenote: this circuit uses BC109 which is ancient. It will work just as well with more "modern" NPNs like BC549 or the not so modern but very cheap 2N2222.