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The methods work because radioactive elements are unstable, and they are always trying to move to a more stable state. This process by which an unstable atomic nucleus loses energy by releasing radiation is called radioactive decay.
The thing that makes this decay process so valuable for determining the age of an object is that each radioactive isotope decays at its own fixed rate, which is expressed in terms of its half-life.
Because plants use carbon dioxide for photosynthesis, this isotope ends up inside the plant, and because animals eat plants, they get some as well.
For example, uranium-lead dating can be used to find the age of a uranium-containing mineral.
It works because we know the fixed radioactive decay rates of uranium-238, which decays to lead-206, and for uranium-235, which decays to lead-207.
For example, how do we know that the Iceman, whose frozen body was chipped out of glacial ice in 1991, is 5,300 years old?
Well, we know this because samples of his bones and hair and even his grass boots and leather belongings were subjected to radiocarbon dating.
Carbon-14 is continually being created in the atmosphere due to the action of cosmic rays on nitrogen in the air.