A digital multimeter (DMM) measures voltage, current, and resistance by using different measurement circuits within the device.
Voltage measurement: A DMM measures voltage by connecting the voltage-sensing circuit across the two points in the circuit whose voltage difference is to be measured. This circuit converts the voltage difference into a proportional current, which is then amplified and digitized for display.
Current measurement: For current measurement, the DMM acts as a load in the circuit and measures the voltage drop across a shunt resistor. The magnitude of the current is proportional to the voltage drop, which is then amplified and digitized.
Resistance measurement: A DMM measures resistance by applying a known voltage across the device being tested and measuring the resulting current. The resistance is calculated as the ratio of the voltage to the current and displayed in ohms.
Note that some DMMs also measure other quantities such as capacitance, inductance, and continuity.