There is considerable evidence that biodiversity promotes multiple ecosystem functions (multifunctionality), thus ensuring the delivery of ecosystem services important for human well-being. However, the mechanisms underlying this relationship are poorly understood, especially in natural ecosystems. We develop a novel approach to partition biodiversity effects on multifunctionality into three mechanisms and apply this to European forest data. We show that throughout Europe, tree diversity is positively related with multifunctionality when moderate levels of functioning are required, but negatively when very high function levels are desired. For two well-known mechanisms, 'complementarity' and 'selection', we detect only minor effects on multifunctionality. Instead a third, so far overlooked mechanism, the 'jack-of-all-trades' effect, caused by the averaging of individual species effects on function, drives observed patterns. Simulations demonstrate that jack-of-all-trades effects occur whenever species effects on different functions are not perfectly correlated, meaning they may contribute to diversity-multifunctionality relationships in many of the world's ecosystems.