This probably enabled humans to drink more milk from their mothers for longer time - I really have no idea why the mutation came along and spread - could have been completely random, or maybe some humans did indeed begin to drink milk from other animals.
There's actually pretty strong evidence that adult lactose tolerance spread because of bovine milk in the diet. For example, cattle were domesticated in northeast Africa about 9000 years ago, and the mutation that originated there for adult lactose tolerance started becoming common about 7000 years ago:
http://www.newsdesk.umd.edu/scitech/release.cfm?ArticleID=1376This was after the paleolithic, in the early neolithic period, and it's likely that the alternative to drinking milk for these populations was starving. In the same time period, agriculture was spreading, human populations were booming, and people were becoming smaller and less healthy, etc.
The north European mutation for adult lactose tolerance is a different mutation with the same effect, and originates in another area where cattle are thought to have been domesticated early.
The fact that multiple mutations happened so recently with this effect has an interesting reason. They are all in the same region of chromosome 2. This region of chromosome 2 is a small distance upstream of the genes coding for lactose digestion, and is likely responsible for turning lactose tolerance off after infancy, so any mutation that damages this region allows the gene for lactose tolerance to stay on into adulthood.
As you say, that doesn't mean we're suited for utilizing all the other ingredients in cow's milk effectively. That would likely require the development of entirely new genes, not just a point mutation to turn one gene off, and entirely new genes made up of many base pairs would take a lot longer than 10,000 years to develop.