Question: Why are eels pests? How do they destroy the environment?

Eels generally are not pests. They like most other species fulfil very important roles within their respective habitats. They would be considered a pest if:
1) An exotic (overseas) eel species escaped or were released into Australia’s waterways.
2) Local species had an impact on some kind of commercial venture (e.g. aquaculture)
3) They had some other negative impact on humans.

I am not aware of any exotic species of eels in Australia, nor where there is any kind of commercial or human impact – however that doesn’t mean it isn’t occurring. If you can provide some more information I’ll look at it further.