What does independent mean here? Not influenced by a previous writing system? Does this imply that all other writing systems have a singular source?
It implies that most of the written languages we're aware of have been based on preexisting character sets, with very few root character sets. This is one of those few root sets.
https://en.wikipedia.org/wiki/History_of_writing#Inventions_of_writing
According to the article, experts currently believe that writing was invented independently at least four times:
There is still debate around Rongorongo script and Vinča symbols, since no is sure if they are actually writing systems.
A very very good and tragic podcast about Easter island, where this is also mentioned. The whole series is excellent.
Fall of Civilizations Podcast - 6. Easter Island Where Giants Walked
Such an amazing series! I love that there are content creators who make documentaries that rival mainstream production companies in terms of quality and their content is practically free.