Living organisms preferentially metabolise C-12 over C-13, so their surroundings tend to show a depletion of C-12. There's a similar change in ratio when organisms metabolise sulfur.
This is a fascinating find as it pushes signs of life back to a point within 400 My of the end of the Late Heavy Bombardment which would have plastered the whole planet with massive craters - but which also brought a sizeable amount of the Earth's water.
There are practically no sedimentary rocks of this age left in the world, so this might be the best we can get for now. Although there is a chance fossils from an older rock could be included in a younger conglomerate or breccia. But pretty much every rock of this age has been through at least one phase of metamorphosis which tends to erase fossils as the rocks get cooked.
There is some evidence of life in even older rocks (3.8 Gy) from the Isua Complex in Western Greenland. Some sequences contain traces of graphite enriched C-12 suggestive that they originally came from sediments including living organisms. However, the Isua rocks are in a greenstone belt which has been heavily metamorphosed into gneiss and they've lost their original sedimentary features and any fossils have been baked away.