A dystopian society is the opposite of a utopian society. A utopia is a world where everything is ideal and all the citizens are happy with the way their world functions. Dystopian societies are very common in science fiction and these societies are usually born through cataclysmic events (whether environmental or catastrophic) which set up the dehumanization usually overseen by a totalitarian government.
So, your question is how this type of society is set up in The Day Of The Triffids. The answer is partly in the title: triffids. Triffids are fictitious carnivorous plants believed to be secretly bio-engineered by the Russians. In the novel, triffids have poisonous stings which the plants use to kill human beings. Triffid plants are farmed for their oil, which makes good feed for cattle. Farmers have to use protective gear while dealing with the plants due to their poisonous stings. So, how did these plants come to contribute to the dystopian society of the novel? Conveniently, the story tells us that as Earth passes through some comet debris, the ultra-bright green flashes produced by meteors lead to the blinding of people who look upon the brilliant display. Of course, the triffids take it upon themselves to take full advantage of this situation; they set about killing as many blinded humans as they can find. Life becomes difficult and economic activity is halted. People must depend upon any sighted individuals they can find in order to forage for food. A week after the comet catastrophe, a plague descends on the whole world, further incapacitating the inhabitants of Earth. Those lucky few who have managed to escape being blinded by the green flashes eventually start a new colony on the Isle of Wight.
So, you can see that the triffids, comet debris and the plague all play their parts in setting up a dystopian environment in the novel.
For your own enjoyment, you might like this website which explains all about the fictitious triffids and also includes chapter summaries. Thanks for the question!