Apologies if I'm repeating what others have said. This is my understanding of the reasoning behind the introduction of the sunday show:
Prior to the first sunday show in series 5, the main show got very good ratings, but the results show didn't. This was for 2 reasons: it was too late in the day for young children and people who liked to go out on a saturday, and it clashed with the X Factor results show.
DWTS and American Idol both had split main/results show (on consecutive weeknights) which the Beeb and ITV observed with interest. Both shows had very good ratings.
Strictly then decided to launch a sunday results show in 2007, and this was launched with the dance off, introduced for a multitude of reasons including as extra padding for the new sunday show. Unfortunately this could not be live as Len and Bruno had to fly to LA on sunday morning to arrive in time for DWTS which is filmed and broadcast live on mondays/tuesdays.
The newly launched sunday show was very successful indeed. ITV watched with interest, and the following year (2008) decided to give Dancing on Ice a sunday night try-out - traditionally the biggest shows fared best on saturday nights. This move was very successful and sunday night officially became the new saturday night, TV wise.
Much to the Beeb's annoyance, ITV decided that there was enough evidence to support a successful live sunday night X Factor result show, which they did in 2009, leaving the later part of saturday nights free.
Strictly then decided, in full-on ratings war fashion, to move the 2009 results show back to saturday, hoping that with TXF results show now on sunday, the Strictly result show ratings will increase. However, as there was the real possibility that viewers won't switch back to Strictly from TXF, a decision was made to broadcast Strictly and its results as one long show, and we all know how that went down...
In the end, in 2010 Strictly backed down and returned to their not-live sunday show, and it looks like it's going nowhere as the ratings are very healthy