Briefly, they're wrong. I responded in detail above.

You are correct, what we're burning as fossil fuels is largely the remains of millions of years of vegetative and microbial life, altered due to heat, pressure, and time. Millions of years of time.

All that carbon making up those organisms was fixed from the atmosphere. While biological functions have been busy fixing CO2, volcanoes, the Earth's mantle, and even some geochemical processes release CO2. If not for biological fixation, the atmosphere's CO2 content would be higher.