The NGOs find ways to route the received money back to the politicians in the form of campaign donations, or having a politician's friend/relative being an executive at the NGO.
The NGOs also subcontract to other NGOs, who take their cut, and eventually just a trickle winds up going to the purpose of giving money to the NGO.
The first part sounds like it's US-specific; campaign donations are less of a thing, and more strongly controlled, in Europe. The second could happen here too, though, and probably does.
I was indeed referring to the US. I don't know much about corruption in other countries.