The left wing media likes him largely because he's closer to their views than almost anybody else in the Liberal Party.
A lot of Liberal Party people think that Turnbull would have a better chance of leading the party to victory at the next election. I have my doubts about that. While he is more popular than Tony Abbott, I think a lot of voters who prefer Turnbull to Abbott prefer Shorten (the leader of the Labor Party) to Turnbull. There is also the risk that the Coalition with the National Party (which is the cornerstone of conservativism in Australia - the major conservative parties have typically been in coalition since 1923, and are generally treated as a block) would be at risk, as Turnbull is generally a supporter of 'climate change legislation' and the Nationals might well leave the Coalition unless he agreed to shelve any ideas he had on that score. I'm inclined to think he would to preserve the Coalition, but it isn't certain. At the moment, the Liberals could govern without the Nationals - but in a close election, the Nationals are crucial - and once the Nationals are spurned, would they come back?
Why is Abbott becoming so unpopular?
The best way for the right party to lose is to place at its head a left-winger (see Willard, McCain in the US).