Fossils are remains of plants, animals, or other organisms that have been preserved in nature such as bones which are sometimes found buried in caves or other similar sites. Fossil records that are well-preserved reveal a lot of information about the past, particularly the environment of the organisms, their interaction with other organisms within their era, and even their relationship with other organisms to which they may be related - whether their ancestors or descendants.
In the study of evolution, fossil records become relevant because, by identifying fossils and using dating techniques to build a timeline of species, ancestry can be established at best, or chronology at worse. With DNA sequencing and carbon-dating, direct relationship between species with available fossil records can be established. This is the ideal case which directly supports evolution. Meanwhile, if no high quality and specific information is available, the existence of fossils, and the specific time point/era in which the organisms existed, will allow for the generation of a timeline of species, which can help a lot in validating evolutionary hypotheses, or generating new ones.
For instance, fossil records can reveal transitional forms between related species. The ancestor of the gray whale had nostrils in front of its skull, while the modern gray whale has its on top. Fossil records have been identified where a related, potentially transitional form, has nostrils in the middle of the skull. Thus, a less coarse transition can be established, and this is made possible by fossil records.
Homologies can also be established. Homology is the degree of similarity between similar structures in organisms. Related organisms will have more similar structures than more diverse organisms -- although the function can complicate things, and since we possibly have not encountered organisms whose fossils are being studied, it may be difficult to be conclusive with regards to function, but structure identification is more definitive.
Fossil records have allowed for identification of similarities and relationships between mammals, birds, and other organisms in the modern world and their ancestors. Another example would be fossil records of ancestors of humans. There are available fossil records that trace evolution of humans by the development of certain features that distinguish humans from their primate cousins - like opposable thumbs and bipedalism. Fossil records also then help in creating ancestral/evolutionary trees. Gaps or missing links are usually filled in once fossil records are found that satisfy the hypotheses.
In general, fossil records support the theory of evolution because it allows us to trace the past to the present.