Sedimentary Rock
Winds and water move earth. Soil and rock are eroded from some areas and redeposited in others. Volcanos, floods and other natural phenomenon continualy change the surface of the earth. During this process organic matter as small as a tiny insect or as large as a dinosaur pelvis get covered in sediment. They harden with pressure and age. Where sedimentary rock is excavated by nature or man, layers are once again exposed. The contents of each layer tell us things about the period of time that coincided with that sedimentary layer.
Organic Matter
Inorganic matter can provide useful information, like the approximate date of a volcanic eruption. But organic matter is of much greater value. For example, an animal skeleton might be found with ingested plant seeds or other small animals which could tell a more comprehensive story. Also, organic matter can be more readily tested using methods like radio-carbon dating.
Archeology and Geology
Fossil records and the record as a whole are of particular interest to both geologists and archeologists. Geologists are primarily interested in natural history. For example, they are interested in the collision of tectonic plates, eruption of volcanoes, and natural dredging of river mouths. Archeologists are interested in discovering evidence of humans, animals and plants.
Ancient Narratives
Anthropologists don't specifically study fossil records, but they use them to help study civilizations. For anthropologists, creating an accurate historical narrative often uses fossil records to triangulate or infer information. So, for example, if an anthropologist has a belief derived from one point of view -- say the stories of a native tribe -- they could reference that set of information with fossil records to confirm or more accurately understand the culture they're studying.