Take for example this simple circuit:
The volatage across $R_1$ is $U_1 =V \cdot \frac{R_1}{R_1+R_2}$ and the voltage across $R_2$ is $U_2 =V \cdot \frac{R_2}{R_1+R_2}$. Now suppose you want to measure these two voltages with a voltmeter, but your voltmeter has a really low resistance $R_v$. If you put the voltmeter parallel to $R_1$ to measure the voltage $U_1$, you will actually almost shortcut $R_1$ so the voltage that you will measure $ U_{meas}$ will be smaller than $U_1$.
More precisely, the resistor $R_1$ and the resistance of the voltmeter $R_v$ in parallel have the resistance $$R_{tot}=\frac{R_1 \cdot R_v}{R_1+R_v}$$ so the voltage that you will measure is actually $$U_{meas} = V \cdot \frac{R_{tot}}{R_{tot}+R_2} \neq U_1$$
But if $R_v$ gets really big $$R_{tot}=\lim_{R_v \to \infty} \frac{R_1 \cdot R_v}{R_1+R_v} = \lim_{R_v \to \infty} \frac{R_1 \cdot R_v}{R_v} = R_1$$
So the measured voltage in this case will be $$U_{meas} = V \cdot \frac{R_{tot}}{R_{tot}+R_2} = V \cdot \frac{R_{1}}{R_{1}+R_2} = U_1$$
For the second part of your question, yes a voltmeter use the tiny amount of current flowing trough to measure the voltage. You might know that a current induces a magnetic field, so the greater this tiny current is, the greater this magnetic field will be. We can then put a magnet next to it and measure the force on the wire and deduce the current from it, which then allows us to know the voltage using Ohms law.
