Step 1: Meaning of the two terms.
Free electrons are the detached valence electrons of a metal that wander through the lattice at high random (thermal) speeds but with zero net displacement. Drift velocity is the tiny net forward speed, of the order of \(10^{-4}\) m/s, that these electrons pick up along the wire once a battery sets up a field inside it.
Step 2: Start from current density.
Define current density \(j = I/A\). In one relaxation time an electron gains drift speed \(v_d = \dfrac{eE\tau}{m}\), so the current density is \(j = n e v_d = \dfrac{ne^2\tau}{m}E\).
Step 3: Recognise conductivity.
Write \(j = \sigma E\) where \(\sigma = \dfrac{ne^2\tau}{m}\) is the electrical conductivity. Since \(n\), \(e\), \(m\) and \(\tau\) are fixed for a given metal at a fixed temperature, \(\sigma\) is a constant. This is the microscopic form of Ohm's law, \(j = \sigma E\).
Step 4: Convert to the everyday form.
Put \(j = I/A\) and \(E = V/l\): \(\dfrac{I}{A} = \sigma\dfrac{V}{l}\), which gives \(V = \dfrac{l}{\sigma A}\,I\).
Step 5: Identify resistance.
The bracket \(\dfrac{l}{\sigma A}\) is a constant for the given wire, and is its resistance \(R = \dfrac{l}{\sigma A} = \dfrac{ml}{ne^2\tau A}\). Hence \(V = IR\) with \(R\) constant, so current is proportional to voltage, which is Ohm's law.
\[\boxed{j=\sigma E \ \Rightarrow\ V = IR}\]