Browning is alluding here to the philosophy of Emanuel Swedenborg, a mystical Christian philosopher, who propounded the doctrine of correspondence:
Moreover, there is no one thing existing in the created world, which has not correspondence with the things existing in the spiritual world, and which does not thereby, in its manner and measure, represent somewhat in the Lord’s kingdom; hence are derived the existence and subsistence of all things.
Emanual Swedenborg (c. 1752). Arcana Cœlestia, volume IV, p. 57. London: Swedenborg Society (1879).
We know that Browning was familiar with Swedenborg as she mentions him many times in her letters, for example:
I don’t think, if you will allow of my saying so, that you apprehend Swedenborg’s meaning very accurately always. If Swedenborg saw sin and danger in certain communications, for instance, why did he consider it privilege on his own part to live in the world of spirits as he did. True, he spoke of ‘danger,’ but it was to those who, themselves weak and unclean, did not hold ‘by the Lord.’ He distinctly said that in the first unfallen churches there was incessant communion, and that the ‘new church,’ as it grew, would approximate more and more to that earlier condition. There is a distinct prospect given in Swedenborg of an increasing aptitude in the bodies and souls of men towards communication with the Disembodied.
Elizabeth Barrett Browning (c. January 1861). Letter to E[uphrasia] F[anny] Haworth. In Frederic G. Kenyon, ed. (1897). The Letters of Elizabeth Barrett Browning, volume 2, pp. 424–425. London: Smith, Elder.
So in the passage from Aurora Leigh the correspondence is that the bloom and death of flowers symbolizes our own journey through life to the spirit-world. Is there a particular passage in Swedenborg that that Browning is alluding to? In Swedenborg’s scheme of correspondence he takes the blossoming of flowers as a symbol of spiritual growth, for example in his commentary on Genesis 40:10:
The man who is born anew, that is, who is regenerated by the Lord, is especially called a heaven; for in this case lie is implanted in good and truth Divine which is from the Lord, consequently he is implanted in heaven; for the man who is re-born, in like manner as a tree, begins from seed; wherefore in the Word seed signifies the truth which is from good: also in like manner as a tree, he first produces leaves, next flowers, and finally fruit: for he first produces such things as are of intelligence, which also in the Word are signified by leaves, next such things as are of wisdom, which are signified by flowers, and finally such things as are of life, namely, the goods of love and charity in act, which in the Word are signified by fruits.
Emanual Swedenborg (c. 1752). Arcana Cœlestia, volume VII, p. 92. London: Swedenborg Society (1879).
I searched the Swedenborg corpus for “flower” and didn’t find anything that quite matches Browning’s correspondence in Aurora Leigh, so I conclude that it does not come from Swedenborg, but is rather a Swedenborgian presentation of the metaphor of the flower’s short life, of which there are many examples, for instance:
This carol they began that hour,
With a hey, and a ho, and a hey nonino,
How that a life was but a flower,
In the spring time, the only pretty ring time,
When birds do sing, hey ding a ding, ding.
Sweet lovers love the spring.
William Shakespeare (c. 1599). As You Like It, act V, scene 3. Project Gutenberg.