Every follower of Bahá’u’lláh knows well that the purpose of His Revelation is to bring into being a new creation. No sooner had “ the First Call gone forth from His lips than the whole creation was revolutionized, and all that are in the heavens and all that are on earth were stirred to the depths.” The individual, the institutions, and the community—the three protagonists in the Divine Plan—are being shaped under the direct influence of His Revelation, and a new conception of each, appropriate for a humanity that has come of age , is emerging.
No sooner had the First Call gone forth from His lips than the whole creation was revolutionized
"The whole creation was revolutionized" at the same instant (no sooner than) "the First Call had gone forth from His lips". This is not the typical word order, so it can be confusing.
all that are in the heavens and all that are on earth were stirred to the depths.
Stir can mean "to affect" and is typically used to describe affecting a state where the one being "stirred" is having difficultly controlling him/her/itself. It's usually followed by the word "up" but this doesn't have to be the case (unless you are stirring up trouble.)
Example: She stirred up within me a longing desire for affection.
The question remains - what particular aspect of "all in theheavens and earth" are affected? - the text doesn't say yet, other than being "revolutionized."
"To the depths" means thoroughly, completely, or as much as possible - if you've heard the saying "X to the core" - as in rocked to the core, it's similar.
"To have come of age" is an expression that means "to mature/become mature" or "to grow up."