The clue to this passage is in the word 'decaying', supplemented by the use of similar words such as 'stale', 'dim', and 'damp'. These  offices appear dark, closed off, musty, and the men within are presented as unappealing, 'grey-faced' creatures trapped in an unhealthy environment.

In short, this description serves to paint a negative picture of the everyday workings of the legal system; it appears backward, out of touch. It is true that these particular offices are not concerned with the workings of the judiciary at the highest level, but such a description does sound a rather alarming note, and helps to reinforce the sense that all is not well in the legal system - a fact which is of course borne out by Tom Robinson's trial. The justice system may be impartial in theory, and Atticus, at least, does keep faith with it, but Robinson's trial shows that, in practice, the system is deeply flawed. Atticus can only hope that in future it will improve, but Robinson's trial shows that it still has a long way to go - that in practice, it remains backward and unenlightened. 

Atticus had used every tool available to free men to save Tom Robinson, but in the secret court of men's hearts, Atticus had no case. Tom was a dead man the minute Mayella Ewell opened her mouth and screamed. (chapter 25)

This quote is a stark realisation of the truth: that, no matter what the elevated principles behind the justice system may be, it still, in the end, cannot get past ingrained social and individual prejudices - 'the secret court of men's hearts' - which ensure that Tom Robinson doesn't really stand a chance from the first. 

Lee's picture of the decaying, musty old legal offices is reminiscent of Charles Dickens's unflattering portraits of the nineteenth-century legal system in England, in novels like Bleak House. Dickens, like Lee in To Kill A Mockingbird, was concerned to show that the justice system of his day was not really fit for purpose.