A new decade and a new Doctor!
There are many factors that could contribute to Season 7's noticeable boost in viewers. For a start, it's in colour, although the move to colour of course would have affected all TV programmes, so in itself would not be a reason for viewers to tune in exclusively to Doctor Who. Plus it has to be remembered that although the show was broadcast in colour (and some viewers might have tuned in specifically for that reason), the majority of UK homes did not yet have a colour set.
However the combination of a new actor playing the Doctor, and the fact that it hasn't been on TV since 21st June 1969 might have been a major factor in enticing viewers back. Since the first episode of the Doctor's debut adventure "Spearhead From Space" was broadcast on the 3rd January 1970, this makes the break between seasons - at 27 weeks - by far the longest run of Doctor-less Saturdays since the show started.
It would appear that the less a show is seen on TV, the more popular it is when it's on. This phenomenon can be seen, for example, when "Only Fools And Horses" and "Jonathan Creek" broadcast their infrequent "Specials" and both enjoy higher viewing figures than they ever did during their weekly runs. And of course the same effect has been seen more recently with Doctor Who's own 2009 year of "Specials", which regularly beat the seasons on either side of it by 2 million viewers.
Regardless of the reason, Season 7 kicks off with a solid 8 million viewers, remaining above 7 million until part-way through the penultimate story "The Ambassadors of Death" when viewers begin to gradually slip. The lowest is about 4.8 million during the 3rd episode of "Inferno", with the season ending in June at just over 5 million. One can speculate that this is down to the novelty of the new Doctor wearing off, but it might just as well be due to external factors, and it would certainly be interesting to know what was being shown in the same time slot on ITV from April onwards.