Not quite sure; the area of South Hampshire has had strong Lib Dem support for a couple of decades
I think it started in the 80s, which was before I moved here. Whether it came from the parts of Labour that split to form the SDP, or from the Liberals, I don't know; but, according to this, the Alliance (as it then was) took control of Eastleigh council in 1987. I would guess the anti-Tory vote coalesced around them - being the south of England, but not in a large city, Labour would not have been that strong, I think. In 1994, they took the parliamentary seat in a by-election; in 1997, they took the neighbouring Winchester seat in the general election (where they've swapped control with the Tories over the past 20 years) , and in 2000, the Romsey seat in another by-election (part of which is in the Eastleigh council area).
They seem to do quite well running the councils (for instance, they were one of the best in the country for recycling for several years, though others have now caught up), and so they built a good local organisation. They did lose the Romsey and Winchester seats in 2010 to the Tories (which I think will stay Conservative this time) - perhaps that shows the "we're not the Tories" effect had worn off after Labour being in government for 13 years), but they held off UKIP in Eastleigh in a 2013 by-election (the Lib Dem MP Chris Huhne had been convicted of lying to the police about a driving offence).
I think Labour has tended to concentrate local effort on neighbouring Southampton, a larger city where they have generally held both Westminster seats recently.