I did a quick work up of Steen's PPG through the years until his 33 goal season:
As you can see, it's a pretty steady career progression. The dips from his rookie year really more attributable to his PP time being cut in half.
(To add insult to injury, his last year before being traded is pretty remarkable. 42 points in 76 games despite only 131 PP minutes and 225 minutes killing penalties. His most common linemates that season were Matt Stajan and Boyd Devereaux)
What changed, then, in St. Louis? Well, aside from his first year there, the low on the above chart, they increased his PP/5v5 ice time by taking him off the PK(under a minute a game in 2010-2011) and put him back on a #1 PP unit.
Steen goes from being someone who doesn't score much to someone who, over the course of four years, scores at essentially a 60 points/82 games rate. That's it. No inexplicable growth. Just more PP/5v5 ice time.
But then we have the second jump, the "breakout year". What caused that? Again, it's really just a change in circumstances. In the four years I refer to above Steen's most common linemate is our old friend Jay McClement. So it's pretty fair to say that his point totals were being dragged down(he also saw a fair amount of time with guys like BJ Crombeen). Near the end of that stretch he started getting time with David Backes and things clicked. He got bumped up to the top line. McClement and Crombeen were replaced by Backes and Oshie, Polak and Carlo Colaiacovo replaced by Pietrangelo and Bouwmeester. His PPG jumped by about .2 a game.
So aside from what I posted above about a rookie scoring .6 ppg moving into the .73 range being a fairly normal progression, it looks like his career was a generally pretty straight shot upwards of what he was capable of, the only thing that changed were his circumstances.
Which, for the record, jives with my memories of the reaction to the trade here. People were upset about the trade because we knew how talented Steen was. There's nothing inexplicable or hard to forecast about his big PPG jump. He just finally got put in the right situation to flourish offensively.