meaning-in-context
His two attempts to break away from his own tradition were signal failures: "Turtle Doves alarmed by Sparrow-hawk" and "Wolves on the Roman Campagna" came back to his studio in the guise of abominable heresies, and Eshley climbed back into grace and the public gaze with "A Shaded Nook where Drowsy Milkers Dream."
quoted from The Stalled Ox
What does climbed back into grace mean?
In American English, we sometimes say up north, down south, out west, and back east when referring to directions. The words up, down, out, and back do not have any special meaning in this context, but are often used as a matter of convention. Additionally, the Eastern United States is sometimes referred to as back east.
According to this Ngram, it appears that this usage became more common in the latter half of the 19th century, began to decline between 1940 and 1970, and then became more widely used than before:
To dish something back is slang for "to speak to someone in a manner similar to their manner when speaking to you".
If someone speaks rudely to you and you speak rudely back, you are dishing it back at them. Or if someone use a lot of jargon when speaking to you and you throw a bunch of jargon back at them, you're "dishing it back".
That which is "dished back" is usually tinged with some form of unpleasantness or strife. If someone praises you and you praise them back, it would be an ironic/humorous usage to say that you were "dishing it back" at them.
Best Answer
To be in good graces with someone means they have a good opinion of you; to be in their good favour (Link).
The best way to explain 'climbing back into grace' in this context is that the subject of your quote, Eshley, possibily had a bad response to their previous works and this new one had a good response from the public and restored their opinion of him/her. They essentially climbed back into good graces.
Edit - There is also the more literal meaning along the same lines: to fall from grace is to get on the wrong side of the people (or god). So climbing back into grace has the same meaning as the above. (Link)