The Zeitgeist is the spirit of a generation or a period of time. For example, during the 18th century, with the spread of the enlightenment, there was a shift towards reason, individualism and skepticism of tradition. This would be the Zeitgeist of that period.
Less than one century later, Nietzsche saw the shift in thought and belief that was brought upon us by science and foresaw what was to come. He writes

God is dead! God remains dead! And we have killed him! How shall we console ourselves, the most murderous of all murderers? The holiest and the mightiest that the world has hitherto possessed, has bled to death under our knife, – who will wipe the blood from us? With what water could we cleanse ourselves? What lustrums, what sacred games shall we have to devise? Is not the magnitude of this deed too great for us? Shall we not ourselves have to become Gods, merely to seem worthy of it? There never was a greater event, – and on account of it, all who are born after us belong to a higher history than any history hitherto!” (Nietzsche, The Gay Science, aphorism 125, 1882)

In the same aphorism he proceeds to write

I come too early,” he then said, “I am not yet at the right time. This prodigious event is still on its way, and is travelling, – it has not yet reached men’s ears. Lightning and thunder need time, the light of the stars needs time, deeds need time, even after they are done, to be seen and heard. This deed is as yet further from them than the furthest star, – and yet they have done it!” (Nietzsche, The Gay Science, aphorism 125, 1882)

We only need to jump forward one century to be in the modern age. We have computers, we have heated homes, we have readily available food, instant communication, social media, but does the majority of the population not feel empty? Perhaps this event is still on its way, and most people did not find a way to deal with the loss of meaning that the end of religious fate brought upon us. Nietzsche divides the population in two groups, the Übermensch, that is basically the superior man that is able to create values for himself (see the metaphor of the three metamorphoses of the spirit in Thus spoke Zarathustra for more) and the last man, which is basically the low man that just seeks comfort and has no higher purpose. In this beautiful passage Nietzsche describes the last man, read it for yourself, it is truly beautiful in my opinion.

It is time for man to fix his goal. It is time for man to plant the seed of his highest hope. His soil is still rich enough for it. But that soil will one day be poor and exhausted, and no lofty tree will any longer be able to grow there. Alas! there comes the time when man will no longer launch the arrow of his longing beyond man — and the string of his bow will have unlearned to whiz! I tell you: one must still have chaos in oneself, to give birth to a dancing star. I tell you: you have still chaos in yourselves. Alas! There comes the time when man will no longer give birth to any star. Alas! There comes the time of the most despicable man, who can no longer despise himself.
Lo! I show you the Last Man.
“What is love? What is creation? What is longing? What is a star?” — so asks the Last Man, and blinks.
The earth has become small, and on it hops the Last Man, who makes everything small. His species is ineradicable as the flea; the Last Man lives longest.
“We have discovered happiness” — say the Last Men, and they blink.
They have left the regions where it is hard to live; for they need warmth. One still loves one’s neighbor and rubs against him; for one needs warmth.
Turning ill and being distrustful, they consider sinful: they walk warily. He is a fool who still stumbles over stones or men!
A little poison now and then: that makes for pleasant dreams. And much poison at the end for a pleasant death.
One still works, for work is a pastime. But one is careful lest the pastime should hurt one.
One no longer becomes poor or rich; both are too burdensome. Who still wants to rule? Who still wants to obey? Both are too burdensome.
No shepherd, and one herd! Everyone wants the same; everyone is the same: he who feels differently goes voluntarily into the madhouse.
“Formerly all the world was insane,” — say the subtlest of them, and they blink.
They are clever and know all that has happened: so there is no end to their derision. People still quarrel, but are soon reconciled — otherwise it upsets their stomachs.
They have their little pleasures for the day, and their little pleasures for the night, but they have a regard for health.
“We have discovered happiness,” — say the Last Men, and they blink. (Nietzsche, Thus Spoke Zarathustra, 1883)

Those lines feel so prophetic to me, I can see the men of the 21st century in those lines.
Wake up, go to work, go home, have a drink to relieve the stress caused by your perfect and easy life, go to sleep. Repeat. A pig in a cage on antibiotics (Radiohead, Fitter Happier, 1997).
In a similar fashion Dostoevsky writes:

And, indeed, this is the odd thing that is continually happening: there are continually turning up in life moral and rational persons, sages and lovers of humanity who make it their object to live all their lives as morally and rationally as possible, to be, so to speak, a light to their neighbours simply in order to show them that it is possible to live morally and rationally in this world. And yet we all know that those very people sooner or later have been false to themselves, playing some queer trick, often a most unseemly one. Now I ask you: what can be expected of man since he is a being endowed with strange qualities? Shower upon him every earthly blessing, drown him in a sea of happiness, so that nothing but bubbles of bliss can be seen on the surface; give him economic prosperity, such that he should have nothing else to do but sleep, eat cakes and busy himself with the continuation of his species, and even then out of sheer ingratitude, sheer spite, man would play you some nasty trick. He would even risk his cakes and would deliberately desire the most fatal rubbish, the most uneconomical absurdity, simply to introduce into all this positive good sense his fatal fantastic element. It is just his fantastic dreams, his vulgar folly that he will desire to retain, simply in order to prove to himself–as though that were so necessary– that men still are men and not the keys of a piano, which the laws of nature threaten to control so completely that soon one will be able to desire nothing but by the calendar. And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! (Dostoevsky, Notes from Underground, 1864)

Now, with this short thoughts I do not mean to say that science is bad or that religion is good, nor that comfort is bad. It is however interesting to think about how empty our generation feels, and when one thinks of why, the answer may just be that people now just feel as though they were merely cogs in a machine, having no purpose, living mechanically, with their little pleasures to numb the pain of existence, just waiting to die and be gone forever.

And now I am left wondering, when mankind will have progressed to such an extent that really, all he will need to do is busy himself with the continuation of his species and eat cake, when robots will be able to produce everything and AI will always be ready to answer all your questions, what then? Imagine everything is free and everyone has a good living by doing nothing, (is this not where progress is leading us?) well then what then? We are going to be free to do whatever we want, but we are not going to have anything to do, everything will be boring.

Perhaps we will have enough time to realize that we are just specks of dust on a flying rock in space, that one day we’re going to die and nothing will be left of us and that we should just enjoy the time we have while it lasts. Yet, while I believe this to be true and to some extent relieving (see the video on optimistic nihilism by Kurzgesagt) I am not completely satisfied with this answer. I still think men needs a higher purpose to be able to live without falling pray to depression.