To address why Wandsworth is described as a key target, it's necessary to summarise the electoral system for UK national elections, as well as local elections in England and Wales. Both use a first past the post (i.e. the candidate(s) with a plurality of votes wins) in individual voting areas (parliamentary constituencies at national level, wards an local level). Since in many wards/constituencies one party or another receives a prohibitive percentage of the vote, only a few seats are usually actually in play, and since for local elections the councillors sit on individual councils, this means that there are also relatively few battleground councils.
In general Labour traditionally does better than average in urban areas, and the Conservatives better in rural England, and at this particular point of the local election cycle, most council seats up for re-election were urban. This is why Labour still holds a massive advantage in terms of seats won, despite picking up relatively few net gains this time round.
More subtly, for a number of reasons (including to a greater or lesser extent, Brexit) Labour has generally been gaining votes from the young and those in high social class jobs and areas which voted remain in 2016, while losing votes from older voters, those in lower social class occupations and those who voted leave. The voters from the first group are found across London, but particularly in some west London boroughs like Wandsworth and Kensington and Chelsea. Meanwhile, the second group is disproportionately found in northern and seaside towns.
All together, this made Labour on the attack in the first set of councils and on the defensive on the other. Again, since relatively few councils changed hands (and most of those that did went to the Liberal Democrats, who are still rebuilding from massive losses which occurred after they entered into coalition government with the Conservatives in 2010) neither main party did anything new this election, instead just picking up a bit better support in areas it was to be expected.
As such, the only real winner was the Liberal Democrats, who went from "pushed into the sea" to "doing well in some areas" and the only real loser was UKIP, who went from "the next big thing" to being of negligible influence.