The irony of the statement comes from the fact that a basic presumption - that things will "get better" because they cannot "get worse" - has already been proved false, yet the speaker maintains the same logical position.
Having been wrong about the impossibility of things getting worse, the speaker nonetheless continues to assert that things are now as bad as they could possibly be.
The irony then is in the statement's implied contradictions of logic. The speaker is saying a few things that do not add up and the difficulty that drives the irony comes from the fact that the speaker cannot know where the "bottom" is.
Knowing that things are bad does not mean that things are as bad as they could possibly be. Thus the speaker makes an assertion that seems logical at first, but which loses its integrity as we investigate the assumption.
While it seems logically true that when things are as bad as possible they can only improve, the speaker is only guessing as to whether or not his current state is indeed "as bad as possible". It could be worse. The phrase recognizes this potential and, as a result, gains its irony.
The heart of irony in this quotation lies in the fact that it is self referential, and also that it is composed of an entirely abstract statement. One would characterize the first clause "things gonna get better", as an optimistic statement, but the problematic part of this is that that does not at all exclude the possibility that things might get 'even worse' before they 'get better.'
Following this optimism with the additionally optimistic statement that 'things couldn't get worse' further distances the possibility of 'things getting better' from a concrete reasoning or evidence to support it. The statement is true that things will get better if they 'can't get worse,' but there is no definition given for why they are 'bad', or why they won't 'get worse', thus there is an open potential for things to become both better and worse.
If they had left it at the statement that 'it must get better,' then the situation worsening would not at all exclude the possibility of things getting worse and then better, or simply improving -- but qualifying that they must get better, because they *can't* get worse, is an ironic statement that deconstructs itself. It uses no evidence to make a claim that can not be proven, and through its own optimism (setting the condition that it *will* get better because it *cannot* get worse) means that the statement is only true if the situation improves and there are no possible ways in which it can be interpreted negatively. Given that we cannot account for all possibilities, saying that it has to improve is an optimistic statement of possibility and saying that it can't get worse is simply an opinion -- subject to being true or false.