Concept: Neopentyl halides are primary halides but are exceptionally bulky. The presence of a quaternary carbon ($\text{C}$ atom bonded to four other $\text{C}$ atoms) adjacent to the carbon bearing the halogen prevents standard bimolecular reactions.
Step 1: {The structure of neopentyl halide.}
Neopentyl halide ($\text{(CH}_3)_3\text{CCH}_2\text{X}$) has significant steric hindrance.
Even though the leaving group is on a primary carbon, the back-side approach for $S_N2$ or the removal of a $\beta$-hydrogen for $E2$ is blocked by the bulky tert-butyl group.
Step 2: {Examine the E1 pathway.}
Because the $E2$ mechanism is sterically hindered, the reaction proceeds via the $E1$ mechanism.
This involves the ionization of the halide to form a primary carbocation ($\text{(CH}_3)_3\text{C}-\text{CH}_2^+$).
Step 3: {Consider carbocation rearrangement.}
The resulting primary carbocation undergoes a 1,2-methyl shift to form a much more stable tertiary carbocation, which then loses a proton to form an alkene (2-methylbut-2-ene).
The driving force is the steric hindrance that makes $E2$ impossible.