Here is the picture of the cable from my rear V-brakes:
As you can see it's worn down. Should I replace it now or is it safe to use it for some more time?
Yes, at least one strand has broken. Looks like this happened where the cable was previously under the fastening bolt. Since then, the cable has been let out, maybe to allow for new brake pads?
It would probably last a while before more strands break. But a new cable is only around $5. It's very important to be able to stop your bike reliably, so I would definitely replace it now. It's an easy change to make, cables ends normally slot in fairly easily to the brake levers on V-brakes.
Yes - replace it as soon as convenient. You might go another year or two with it in that state, but its a couple of dollars for a straddle cable.
Check the front one too, and consider replacing both cables.
While you're at it, give all the brake cables an eyeball, and if you find any damage then replace them all. A pair of inners costs under $20.
They're brakes - they have to work when you need them.
Derailer cables operating an indexed derailer wear out much faster than brake cables, and in fact brake cables can go a long time before needing replacement. However, while a broken derailer cable is inconvenient, a broken brake cable is dangerous.
I would replace it right away. Brake cable replacement, including the labor, is cheap--certainly when compared to the cost of an ER visit.