Why does ammeter or voltmeter measures RMS value

RMS measurement, like average and peak, only applies to measuring AC, though it may be superposed to a DC offset.

Measuring RMS values is a bit more expensive than measuring average values, so most multimeters avoid the former. Instead they presume your signal is a sine and measure the average value for the rectified sine or the peak value, after which they apply a conversion factor to find the presumed RMS value.

\$V_{RMS} = 0.71 \times V_{PEAK} = 1.11 \times V_{AVG}\$

For other waveforms than sines this calculated RMS value will be wrong! The ratio \$\dfrac{V_{PEAK}}{V_{RMS}}\$ is known as the signal's crest factor,

and this can be significantly larger than the \$\sqrt{2}\$ value for the sine. If the crest factor is 3 and the multimeter would actually measure peak voltage you would have a 100% error for the calculated RMS value. Usually this error is smaller when the averaged rectified signal is measured instead. We're talking about the form factor then instead of the crest factor.
So the lesson is: be very careful when AC measuring anything else than a sine on those multimeters.

Solution: some more expensive multimeters measure "True RMS".

Just like measuring averages true RMS measurement includes an averaging over a certain period. Only when this period is an exact multiple of the signal frequency this will give the most accurate result. If this time constant is a multiple of 100ms accurate results for 50Hz and 60Hz are possible (5 periods and 6 periods, resp.).
Thomas points out that not all True RMS multimeters can measure AC superposed to DC.

Further reading:
AC Voltage Measurement Errors in Digital Multimeters (Agilent application note)

There are a variety of ways of making the actual measurement.

One of the more traditional is a moving-coil micro-ammeter, consisting of a winding on an iron core that rotates between the poles of a permanent magnet, working against the force of a spring. The meter circuitry and selector connects various resistors to scale the input voltage to a tiny current. A crude, related instrument can be made by winding a few turns of wire across a plastic-body magnetic compass, with the introduced field forming a vector sum with the earth's field and resulting in a new pointer angle.

The next major development consisted of using a high impedance vacuum tube amplifier between the circuit under test and the meter movement, producing the Vacum Tube Volt Meter or VTVM. Later the tube was replaced with a field effect transistor.

The third major development would be replacing the meter movement with an analog to digital converter. This usually consists of a comparator which compares the unknown input against known reference voltages; either sequentially as the reference voltage is changed by a digital to analog converter under a search algorithm, or by charging a capacitor while measuring time, or for high speed applications bu using a number of of comparators and voltage sources acting in parallel to produce a faster answer (though that form is more likely to be found in a high speed instrument such as an oscilloscope than a typical digital multi meter).