In the Assange case the charge is "conspiracy to commit computer intrusion." The US is asking the UK to be a friend and arrest this person and then send them to a place where the laws apply so they can be punished for what they are accusing him of:
"Prosecutors accused Assange of conspiring with Chelsea Manning, a U.S. Army intelligence analyst in Iraq in 2010, to crack a Defense Department computer password for accessing a vast trove of classified U.S. military and diplomatic material that subsequently was disclosed through WikiLeaks."
So just think of it like this: The US, UK and Australia are all "friends." The US thinks someone from Australia who was living in the UK did something that the US defined as bad. The US wants him to face trial for that and the only place to do that is in the US, so can their friends in the UK please send him along.
If Assange was in a country that had no treaties with US, say for example Iran, he could avoid extradition easily, not so easily if there are strong agreements on both parties like UK and US have.
There are reasons why he wasn't in Iran or Afghanistan, one of them is that his life would be in more danger there than in any western court.
If Assange was extradited from Sweden, the country he escaped from, which is still an European country, he had the certainty that he could not be sentenced to the death penalty, that's a non-negotiable option if US ask for extradition from EU. UK will not be EU anymore starting from 1st of February 2020.