The time you are talking about (1900-1940) is called the "age of anxiety" in this text because it was a time in which old values (held since the Enlightenment) were being called into question.  This was especially true after WWI because the war seemed to have proven that human history was not this story in which people made gradual but constant progress towards being more rational and more "civilized."

Bergson's quote seems to be saying that everything that exists changes.  It does not necessarily change for the better; it just changes.  This is relevant to the period you are talking about because when things are constantly changing, uncertainty enters in.  When there is uncertainty, there is anxiety.  Instead of living in a world that was making clear progress towards an ideal, the people of this time were suddenly living in a world where everthing was changing.  This caused them anxiety.

You can say, then, that Bergson's quote sums up what was going on during this time.  Everything was changing and nothing seemed constant and safe.