Well I can't seem to find a full methodology to explain the reason for the apparent contradiction. Furthest right hand column seems to suggest they're assuming a relatively constant GDP per Capita prior to 1500 based on the Malthusian idea that when the GDP as a total drops, and give the majority of pre modern economies is agrarian that generally directly or indirectly relates to changes in food production, the population will drop as people starve, and particularly as children go hungry infant mortality among the poor spikes with kids being more vulnerable to childhood diseases when suffering from malnutrition. In reverse when more land is cultivated or done so more intensely causing the GDP to rise infant mortality rates drop. I can't see any explanation for the 1-14 change therefor in their methodology but i can propose some world events that would likely have resulted in a contraction of the GDP globally.

The big one would likely be the flooding of the yellow river and several of its tributaries. As one of the largest river systems in the world, and in particular forming a lot of the backbone of the Chinese Han Empire. The starvation and displacement caused by this combined with the political unrest surrounding the end of the Western Han dynasty and rise of Wang Mang and his short lived Xin dynasty. Wang didn't have much luck because his economic reforms he attempted to deal with the general social unrest and the like that the flooding and political change over had brought were unsuccessful. He also had to wage a campaign against the Xiongnu. All this meant a lot of the peasants were no doubt displaced and/or suffering from starvation and also on account of the conflicts of this period having to suffer a not insignificant tax burden.

Meanwhile in the middle east the Parthian Empire went through a period of turmoil most of which seemed to be palace coups and assassinations but by 12AD broke out into civil war. After some proxy fighting in Armenia during the first few years AD Phraates V backed down and surrendered Armenia into the Roman sphere of influence. For this in 4AD he suffered a palace coup by his nobility and was forced to flee the country. His successor Orodes III was assassinated two years late. We are told Augustus managed to manoeuvre a Roman educated member of the Parthian Royal house, Vonones, onto the throne following the assassination of king. But for this reason he was deeply unpopular with many Parthian nobles for this Roman influence. At some point between 8 and 12 AD civil war broke out in the Parthian empire as a result that eventually resulted in the rise of king Artabanus II.

The increase in the population may not have taken account for this or they may feel that this period of relative internal peace and prosperity within the Roman Empire was enough to outweigh this and still resulted in a net increase in the global population.