How does a higher range when measuring current and voltage prevent burning of the fuse in multimeters?
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Current range on a typical basic multimeter. The burden resistor values shown are for ease of calculation and assume that at full scale the burden voltage will be 200 mV.

How it works:


Most of these meters use an LCD module with a full-scale reading of 200 mV (actually 199.9 mV). If you exceed the full scale reading they typically display '1   '.
On the 10 A range current flows from the 10 A socket to COM through R5. At 10 A there will be a voltage drop of 100 mV across R5. The meter will read this (and it has a high input impedance so R1, R2, R3 and R4 won't affect that) and will display 1000. The additional contacts of the range select will add the decimal point in the right location so that 10.00 A is read.
Each of the other ranges selects a total burden set to give 200 mV at full scale current.
I have shown a 10 A fuse on the 10 A range. Cheap meters usually omit these as the meter is probably cheaper than the fuse. This could be lethal. Good quality meters are expensive for a reason.


Voltmeters have a very high input impedance and so do not draw significant current and do not require a fuse. For more details on auto-ranging see @RusselMcMahon's answer to https://engineering.stackexchange.com/a/560/3505.




  If I start with the highest range of measuring, 10A, the current will pass through a 10A-fuse (if present) and then through a 10mΩ-resistor. The fuse and the resistor will together prevent damage to the meter or?


The meter will be able to handle up to 10 A on the 10 A range and the fuse gives protection should you exceed that. It should be clear that if you try to measure 10 A on one of the lower ranges that you will overheat the resistors. Even if you didn't burn out the resistor you would be introducing 10, 100 or 1000 times the voltage drop on the circuit you are measuring depending on which range you selected.