Fossils were the first widely recognised evidence for extinction; it’s fairly obvious that nothing like a t-rex is alive today. Once the theory of evolution became established, people began to hypothesise now extinct missing links between similar looking species. Nowadays there are a host of other ways of inferring the existence of now extinct species; unoccupied ecological niches, survival adaptations for absent predators, seed dispersal mechanisms that rely on missing hosts, etc.
We do find evidence of plants and animals in the ground. Usually as some sort of fosil. We are then able to date this evidence. There are various ways of dating things like this and we usually use more then one technique in order to verify the age of something. When we collect these evidence from multiple sites around the world we can tell when a species existed. And you are technically correct that we can not truely know when it became extinct. The only evidence we have of a species going extinct is often the lack of evidence. But given how many archeological sites we have with fosils from all different ages we would expect at least some evidence of the species if it had been around. So we theorize that the reason we do not find any evidence is because they have gone extinct. Sometimes we can also provide an explanation as to why they would go extinct although hard evidence for this can be impossible to find.
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