As evident by the Ref.1, the first step is the reaction of triphenylphosphite and methyl iodide to give methyltriphenoxyphosphonium iodide (See path B of Ref.1). This reaction is conditional and some details of conditions are described in Ref.2. Then, one of the phenolate ion in methyltriphenoxyphosphonium iodide is replaced by neo-pentyloxy group giving away a phenol molecule. The resulting compound, suggested by the original authors of the method (Ref.2), undergoes $\mathrm{S_N}2$ mechanism to give the resulting neo-pentyl iodide in good yield:

However, the suggested $\mathrm{S_N}2$ mechanism leans heavily on the claim that rearrangement does not occur, but in light of the finding of Ref.3 (the presence of more than 6% of t-amyl iodide in the reaction mixture that can only achieved by the rearrangement of neo-pentyl group, loses much of its credibility. Accordingly, to explain the rearrangement, Ref.3 suggests following two paths for the mechanism:

According to Ref.2, $\bf{A}$ and $\bf{B}$ are in equilibrium. The covalent form $\bf{A}$ would undergo “cis elimination” thus giving rise to a cyclic transition state, $\bf{a}$ leading to give pure neo-pentyl iodide (See Path 1). Alternatively, the ionic form $\bf{B}$ would break up to give an ion pair $\bf{b}$, which collapses to neo-pentyl iodide before rearrangement occurs (Path 2). However, when combination of neo-pentyl and iodide ions is delayed, the rearrangement would occur and t-amyl iodide would be formed (Also, according to Ref.3, the material balance has never exceeded 80% and the deficiency may well be due to olefin formation).
References:
- H. N. Rydon (Checked by W. Fuhrer, R. Keese, and A. Eschenmoser), "Alkyl iodides: neopentyl iodide and iodocyclohexane (Propane, 1-iodo-2,2-dimethyl- and Cyclohexane, iodo-)," Organic Syntheses 1971, 51, p.44 (DOI: 10.15227/orgsyn.051.0044); Coll. Vol. 6 (1988), p.830 (4 pages).
- S. R. Landauer and H. N. Rydon, "458. The organic chemistry of phosphorus. Part I. Some new methods for the preparation of alkyl halides," J. Ckem. Soc. 1953, 2224-2234 (DOI: https://doi.org/10.1039/JR9530002224).
- Nathan Kornblum and Don C. Iffland, "Neopentyl Iodide," J. Am. Chem. Soc. 1955, 77(24), 6653–6654 (DOI: https://doi.org/10.1021/ja01629a083).