Explain to someone who knows nothing about fossils the study of invertebrate paleontology and how it scientists understand the history of the earth.

Paleontology is the study of the history of life on Earth as based on fossils. Fossils are the remains of plants, animals, fungi, bacteria, and single-celled living things that have passed away and replaced by some rock material. They're impressions of organisms preserved in rock. Paleontologists use fossil remains to understand different aspects of extinct and living organisms.
Vertebrate paleontology and invertebrate are subfields of paleontology. It is the scientific study of ancient invertebrates. They do this by analyzing invertebrate fossils.
All of this, all this studying helps to know our past. Perhaps our descendants. It helps to know how the animals at whatever period were capable of, how they lived, if they were living good or struggling, perhaps possible death situations that may happen to us and prevention, etc...

Well it helps scientists understand the history of the earth by knowing what might of lived on the earth thousands of years ago.