I just do not understand how he claims to have defeated the old problem.
From my understanding, I do not think (please do correct me otherwise) that Goodman claimed to have "defeated" the old problem. Rather, all he did was add to it, further questioning induction. According to Stanford Encyclopedia of Philosophy (an excellent resource), the Problem of Induction is an ancient one, generally involving people questioning the value of induction through different arguments. I will try to explain what Goodman's new problem of induction is, so that you can have a better grasp of its argument.
What Goodman did was present "a different description of the problem of induction" in his book Fact, Fiction, and Forecast where he questions the validity of assumptions arising from human habit. In his famous grue paradox (detailed logically below), he creates a new adjective, grue, to describe all objects that are blue after time t and green before time t. Given that all emeralds we have seen are green, he asks why we assume they shall continue to be only green; why can they not be grue? The standard scientific response (this is from Wikipedia) is to invoke Occam's razor, and incidentally, this recent question explains the razor.
So, instead of "defeating" the problem of induction, Goodman added more to it. In the SEP, the following example is given from Goodman to illustrate the mental bias involved in induction (the human habit):
Taken by itself, the statement that [a] given object is neither black nor a raven confirms the hypothesis that everything that is not a raven is not black as well as the hypothesis that everything that is not black is not a raven. We tend to ignore the former hypothesis because we know it to be false from abundant other evidence — from all the familiar things that are not ravens but are black.
The SEP also provides a detailed explanation of Goodman's grue paradox:
Suppose that at time t we have observed many emeralds to be green. We thus have evidence statements:
- Emerald a is green,
- Emerald b is green,
And these statements support the generalization:
All emeralds are green
But now define the predicate “grue” to apply to all things observed before t just in case they are green, and to other things just in case they are blue. Then we have also the evidence statements
- Emerald a is grue,
- Emerald b is grue,
and these evidence statements support the hypothesis
All emeralds are grue
Hence the same observations support incompatible hypotheses about emeralds to be observed in the future; that they will be green and that they will be blue.
The SEP inserts a very important footnote, that:
No one thinks that the grue hypothesis is well supported. The paradox makes it clear that there is something wrong with instance confirmation and enumerative induction as initially characterized.
Through his new problem, Goodman is further questioning the validity of induction, since it involves human prejudice/habit, and can result in the grue paradox. Thus, the only difference between the old problem of induction and Goodman's new version is that there is... more of a problem.