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Fossil records provide evidence of natural selection because one can see what species and what type of species (specifically what type of special trait the species had) reproduced and lived for longer.
For example, we have 5 different fossils of the same animal;
1. Lived about 45 years; had long legs that made it run faster - 5 offspring
2. Lived about 8 years; had short and stubby legs, was very slow - 0 offspring
3. Lived about 67 years; had long legs that made it run faster and a good nose - 8 offspring
4. Lived about 40 years; had a good nose and normal legs - 6 offspring
From all of that information you can see that the prefered trait was having longer legs and a good nose survived and reproduced multiple time, you can also see that the organism that had short legs did not survive for long and did not reproduce.