He says the following:
Let's use a multi-asset local volatility model calibrated for each stock on its market smile of maturity $T$ (a one-maturity smile), and with the Brownian motions correlated through a correlation matrix $\rho$
Then there exists a local volatility for each asset such that: (1) the smile of maturity $T$ for each asset is recovered, (2) the resulting multi-asset local volatility price is equal to the Gaussian copula price with a correlation martrix equal to $\rho$ and the marginals calibrated on the $T$-maturity smiles. This is true for any European payoff.
Given a $T$-maturity smile, there exist many different local volatilities calibrated to this single maturity smile. They generate different smiles for shorter maturities. The local volatility that recovers the copula price is the one generated by a Markov-functional model built on the $T$-maturity smile.
He explains it better than me in section 2.10 of his book. Chapter 2 of his book is posted for free on his website: www.lorenzobergomi.com.