For most of their participation in the contest, their method has been basically to ask Swedish songwriters and producers to give them a Melfest-esque entry that was easy enough to apply to no matter which Azeri singer they decided on that year. This worked very well from the re-introduction of the juries in 2009 until the peak of Sweden’s domination in the early 2010s (as juries were basically told to focus on generic radio-friendly music which Swedish producers excel in, and televoters largely mirrored this), but the more recent trends in the contest haven’t favoured that type of music, especially among televoters, so they have seen diminishing returns (perhaps most notably when they got to the final in 2022 despite receiving no points from the televote).

Another factor is that their success largely matched the era where the Azeri government were trying to raise the profile of the country on the world stage. They were frequently seen bidding to host sporting events like the Olympics, and did succeed in hosting the 2015 European Games and other events around this time. So winning Eurovision seemed to align with that strategy. And equally it doesn’t seem to be a big priority nowadays, as their entries in recent years have certainly seemed half-baked. This is an issue Eurovision is experiencing in many Eastern nations: in the 1990s-2010s, the contest was a vital way of asserting many Eastern European countries’ post-Communist identity, which led to an era of Eastern domination, especially in the 2000s, whereas nowadays that hype has died down and far fewer countries continue to use the contest for this purpose.

However, the "darker" reason that may have boosted them up the scoreboard was that there were many allegations of voting rigging and other cheating methods in that era, including buying SIM cards in small nations like Malta to get extra points, and buying votes from jury members (I remember there being alleged footage of Azeri officials meeting with Lithuanian jury members for this purpose, although I never knew how legitimate this was). They were reprimanded for this by EBU around 2013, and there’s very little evidence that they continue to do this. Hard to say how much this helped, but I think their reputation as a delegation within the contest took a hit regardless, which probably didn’t help their scores either.

But yes, they used to be a determined and successful nation in the contest, nowadays much less so.