Here the proper answer:
The vast majority of devices are constant-voltage devices. They need a constant operating voltage to work correctly. Microcontrollers are such devices. When operating, they will try to draw as much current from the power source, as they need, but not more. Much like a simple resistor with an applied voltage will draw a specific amount of current (calculable with Ohm's law), complex devices like microcontrolelrs have something like an overall impedance/resistance, that is depending on their specific operation. So also here the device only draws as much current, as it needs. (I wrote about an overall impedance to emphasize the fundamental principle. No one would really specify an overall impedance for such a device, because this would only be a value calculated by the voltage and current needed, not a real resistance.)
This means, that burning parts are not due to the power supply supplying more amps. They get burnt, because something draw too much power trough them.
Let's look at some examples:
Imagine, that you create a short between 5V and ground on your Uno. A short means, that a high current can flow, since the resistance is very low. Now you are drawing a high current from your 5V pin to the ground pin. The Arduino's 5V pin is connected to a voltage regulator (for if you use Vin) and over a diode to USB. Both parts are only rated for a maximum current (depending on the version of the board between 200mA and 1A). The high current, that you draw, will flow through one of these parts, depending on your input source. This part will get fried, because a way to high current will flow through it, just because you draw the current.
A strong motor, that needs more current, than the board is rated for, would result in a similar problem like the short circuit. Indeed every device, that needs more current, than the board is rated for, might result in frying the mentioned parts.
But what happens, if you draw more power from the power supply itself, than it is rated for? The power supply electronic is similar to the Arduino board. When you draw too much current from the power supply, some parts of it might get fried and stop working. Thus, you should not try to draw more current from it, than it can provide. Often power supplies have a protection circuit, that will either lower or directly cut the voltage, if too much current is drawn. This is for the safety of the supply and should not be used for current limiting your own application normally.
So, you should always calculate, what current you expect to need, and then take a power supply, that can provide more current than that. You should consider a significant headroom, so that you can add more parts, if you want to extent your project. For example, if I calculate 2A for my project, I would use a power supply in the range of 3A.
Addition:
Above I wrote about constant-voltage devices. There are also constant-current devices, for example LEDs. An LED has a specific constant forward voltage. Under that voltage it will not conduct current. Over that voltage, the current rises exponentially. The voltage over the LED will always stay the same, as it was designed. This is the reason, why you need to add a current limiting resistor to a LED. But in the vast of devices, LEDs (or in fact every diode) is an exception.
