I think part of the answer is right in the definition of the current itself, which is given, for example, by Wikipedia:

Existentialism is a branch of philosophy whose main subject of study is a person, his problems, difficulties, and existence in the world around him.

It is impossible not to agree that the present freedom and a certain uncertainty of the development of events in life make existentialism even more necessary than when it was born, when such freedom was not yet widespread. It is vital for a person, especially a young person, to understand (especially if he thinks about it) how to cope with problems, difficulties, and how to come to terms with some things in life.

Thus, J.-P. Sartre and A. Camus, highlighting the ill-being of the world and showing its absurdity, suggest not to lose heart, but to courageously fulfill their human duty, without fear of loss, without bowing under the blows of fate, calmly do their everyday work, when the oppression of reality becomes unbearable, to dare to revolt, eliminating and weakening this oppression.

In addition, in a world where religion is increasingly devalued, it is increasingly difficult for a person to live with the knowledge that there will be nothing after death. It makes life meaningless, doesn't it?

Also existentialism often shows: whatever happens to a person, most often it's their fault. You can not blame others for your troubles and justify yourself at the expense of other people.

And, in addition to the above, it seems to me that a big influence, as well as it has always been, fashion on philosophy and existentialism.

That is, in my opinion, there are two objective reasons for the mass enthusiasm for this current of philosophy?

1) The need for thoughts expressed by representatives of existentialism in their works.

2) The mainstream.

I think part of the answer is right in the definition of the current itself, which is given, for example, by Wikipedia:

Existentialism is a branch of philosophy whose main subject of study is a person, his problems, difficulties, and existence in the world around him.

It is impossible not to agree that the present freedom and a certain uncertainty of the development of events in life make existentialism even more necessary than when it was born, when such freedom was not yet widespread. It is vital for a person, especially a young person, to understand (especially if he thinks about it) how to cope with problems, difficulties, and how to come to terms with some things in life.

Thus, J.-P. Sartre and A. Camus, highlighting the ill-being of the world and showing its absurdity, suggest not to lose heart, but to courageously fulfill their human duty, without fear of loss, without bowing under the blows of fate, calmly do their everyday work, when the oppression of reality becomes unbearable, to dare to revolt, eliminating and weakening this oppression.

In addition, in a world where religion is increasingly devalued, it is increasingly difficult for a person to live with the knowledge that there will be nothing after death. It makes life meaningless, doesn't it?

Also existentialism often shows: whatever happens to a person, most often it's their fault. You can not blame others for your troubles and justify yourself at the expense of other people.

And, in addition to the above, it seems to me that a big influence, as well as it has always been, fashion on philosophy and existentialism.

That is, in my opinion, there are two objective reasons for the mass enthusiasm for this current of philosophy?

1) The need for thoughts expressed by representatives of existentialism in their works.

2) The mainstream.