How does a battery only send the amps amount the device needs, and not more?
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Simply put, the premise of your question is faulty in just one word. A battery does not send current, a circuit pulls it.
Now for the water analogy! (Sorry, because water analogies fall apart eventually, but this one works well enough.)
When you get a drink of water, you aren't obligated to consume the entire water supply, even though it is technically available. The same is true for a circuit, or a simple light bulb or motor. It only pulls what it can use.
But let's say you were a zombie with gaping holes! Now you can drink and drink and drink, but the water just keeps running out onto the floor. So, in a way, this would be similar to a short circuit or a device with a problem; it draws much more current than it should, and might trip a circuit breaker, blow a fuse, or (if those safeties are not in place) cause a fire or other damage.
Check out the other questions I linked in a comment earlier, they'll help you gain a bit more insight:
Ohm's law: E=IR. The circuit has a certain resistance and the battery has a certain voltage.
Now, that said, the battery is not a perfect voltage source. As the current goes up, its ability to produce voltage will hit a plateau and the voltage will sag. But on a theoretical basis with a perfect power supply, Ohm's law is the answer to the question.