Vestigial structures are features on the body that don’t have a purpose in the current form of an organism in a species.
The features start out as important players in the functioning of an organism. As natural selection occurs over time and the population of a species changes, the importance of those features phases out. Eventually, they become useless to the species.
Often, these features will be completely eradicated from a lineage. However, many of these structures stick around the gene pool, appearing within populations without serving any function. These are vestigial.
How do vestigial structures show proof of evolution?
Many of the structures are homologous, or appear the same, in other organisms.
This shows common descent between organisms. Common descent is a theory that says all organisms descend from a common ancestor. Besides analysis of DNA, homologous and vestigial structures give the non-scientific community direct, visible evidence of the evolutionary process.
This recent video from Vox shows modern human vestigiality that you can observe right now on your own body!