That's a nice simple narrative for a complex issue, hence its appeal. Unfortunately it's bollocks. You can tell that by the fact I raised above: Many of the countries where population growth is still strongest are only contributing a tiny proportion per capita to global climate change; while wealthy western countries with much slower population growth (in some cases zero or negative growth, only outweighed by immigration) contribute the lion's share of emissions and those emissions continue to rise.
The country with the largest Co2 emissions is of course China, yet it's the only country in the world that until recently actually had a clear, enforced policy to REDUCE population. That completely contradicts your simplistic explanation, so how can it be so? Because China's been rapidly industrialising to bring millions of people out of poverty, moving people from subsistence agricultural lifestyles to consumerist middle class ones, and producing a huge amount of the unnecessary luxury goods consumed by those same lifestyles in the west.
The country with the largest Co2 emissions per capita is the USA, yet population growth there has been around 0.5% for the past few years. So similarly, can we work out why?
Overpopulation has always been massively skewed towards developing countries. If you want to argue against it, then the argument is about African and Asian families having 8 or 10 children for cultural and economic reasons, not whether someone in the UK has the replacement level, slightly below or lightly above. But if you're talking about climate change it's largely irrelevant anyway, because industrial and lifestyle factors are so vastly different across the world that responsibility for emissions just doesn't map with any accuracy to population growth.
The world is burning because we emit too much Co2. The amount of that emission cause by each individual in western middle classes has grown SO much over recent decades due to economic and lifestyle changes that even if the entire industrialised west had had no population increase at all in that time it would still have grown.
The problem is our overconsumption, and our addiction to economic growth and material wealth. As a group, we need to get a lot poorer (although there are political arguments to be had about how that reduction in material wealth should be distributed).