This is the most concise explanation that I could find on the ETS, from NZ, as they are/were in the same boat:
Where does the money for carbon credits go? - Business News | TVNZ
There are a couple of ways New Zealand can offset the amount which it exceeds its limits: buying carbon credits, or funding an emission reduction project in a developing country, such as planting trees or investing in renewable energy.
Essentially, the money used to purchase carbon credits will go to funding projects that reduce carbon dioxide emissions. This is because as few countries are expected to have emissions below their 1990 levels and therefore will not be able to offer up their surplus to other countries.
Also, the money paid for carbon credits is likely not to be spent in New Zealand.
When you buy something, the money goes to the seller. The seller is the European Union.
When it was a 'carbon tax', the government set the tax and collected the revenue. When it's 'carbon credits' we buy then from outside Australia at whatever price it's selling.