Exponential decay is a commonly observed phenomenon in nature where the rate of decay is directly proportional to the amount of stuff you start with. If you have twice as much stuff, it decays twice as fast, etc.
Another way to say it is that every molecule has an equal and independent chance of decay.
Radioactive decay is often taken as the "standard" model for exponential decay. Any unstable atom has a chance at every moment in time to decay to a more stable state. Other atoms in the universe don't influence the rate, it's just an intrinsic property of each separate atom that it has some chance to decay.
In pharmacology, the exact mechanism depends a bit on the molecule, but let's imagine a drug that's metabolized in the liver. If the drug is in a relatively low concentration (as most drugs are once they reach the bloodstream) as are it's metabolites, then the rate limiting step is the chance of an interaction between a molecule of the drug and whatever enzyme catalyzes the reaction to the metabolite. If the concentration of liver enzymes remains constant, and the low concentration of drug means there is no appreciable saturation of those enzymes, then you can think about this chance interaction as being an independent chance for each molecule of the drug. If you have twice as many molecules of the drug, you have twice as many chances for them to interact with the enzyme for a given time step.
For something filtered by the kidney, you could think exactly the same way: it's the chance of each individual molecule being filtered into the urine. Twice as many molecules means twice as many chances to end up in the urine.
Not all drugs follow this pattern. Few drugs are in high enough concentrations for the drug concentration to matter, but some are. Ethanol, for example. Other drugs have more complicated metabolic fates, induce enzyme activity that influences metabolism, or accumulate as waste products that reduce decay. There are also drugs that partition strongly into other compartments and you might need to consider different routes in which a drug might eventually leave the system.