There's a short article on the Discovery channel page about the longer weaning period of mammoths relative to modern elephants. Breast milk differs from the plant-rich diet of adults by being richer in heavy isotope nitrogen and poorer in 13-C. I couldn't understand from the article what the scientists did exactly, but I gather that as the tusk grows outward, the tips retain the isotopic signature of the early years, and the base of the tusk is added last (see this abstract from the Fisher group). Very likely then they analyzed segments from the tip and base of a relatively complete juvenile tusk for isotopic differences, and found evidence in the tips for six or more years of nursing.
Modern African elephants nurse their young for about five years. It could be that the harsher climate in which mammoths lived would have required prolonged access to high-fat milk for the kiddies.