It means that it will have to change soon. The word 'more' refers to the proportion of the remaining 80 years of this century.
It's now nearly 2020. We have already been consuming meat in an unsustainable way for the first twenty years of this century. We can't continue 'for much more of the 21st' [century] means that we cannot go on like this 'for many more years of this century'. For example, we can't go on cutting down forests for cattle grazing at this rate until for example, the 2070s or the 2080s. We need to change these practices before then - say, by 2030.