Nice question! I will directly begin with the process through which methamphetamine causes damage to neurons, putting in as much details as are known, and adding appropriate citations wherever required.
Methamphetamine (METH) is known to act by increasing concentration of dopamine in brain1. When excess of dopamine is produced, it causes oxidative damage to axon terminals2. This is because direct oxidation of dopamine leading to quinone formation3, iron-catalyzed dopamine metabolism via Fenton reaction4, and metabolism of dopamine by monoamine oxidase-A5 contribute to production of superoxide and hydrogen peroxide.
Methamphetamine is known to increase concentration of glutamate. This (supposedly) happens as: high concentration of methamphetamine D1-mediated striatonigral GABAergic transmission, which in turn activates GABA-A receptors in substantia nigra pars reticulata (SNr), leading to increase in GABAergic nigro-thalamic activity which increases corticostriatal glutamate release6 (since I can't explain all those terms in a single answer). Now, excessive glutamate activates the glutamate-activated NMDA and AMPA receptors which, in turn, increases activity of nitric oxide synthase7. Activation of nitric oxide synthase generates reactive nitrogen species which create an oxidative stress8.
Oxidative stress is further increased because of depletion of antioxidant enzymes due to methamphetamine itself9. Now, the oxidative stress has to be manifested somehow. This is done by lipid peroxidation and protein carbonyl formation10, and also by specific nitration and nitrosylation of proteins important for monoamine synthesis and release, including VMAT-2 and tyrosine & tryptophan hydroxylase7,11,12. Oxidative modification of these proteins restricts their activity and contributes to their degradation, thus playing a major role in neurotoxicity. Researches substantiate the significant contribution of oxidative stress to the neurotoxicity of substituted amphetamines antioxidant treatments have also been shown to be neuroprotective against the damage produced by methamphetamine or methylenedioxymethamphetamine13,14,15.
P.S.: oxidative damage is not the only way through which methamphetamine causes neurotoxicity. Other possible ways include altered metabolism16, damage to mitochondria17, peripheral organ damage18 due to oxidative modification of hepatocellular mitochondrial proteins19, activation of microglia in striatum, cortex and hippocampus20, etc. Also, cannabinoids are known to suppress inflammatory processes and damage during methamphetamine exposure21. But since this is out-of-scope for this question, I won't go into their details.
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10. Gluck MR, Moy LY, Jayatilleke E, Hogan KA, Manzino L, Sonsalla PK. Parallel increases in lipid and protein oxidative markers in several mouse brain regions after methamphetamine treatment. J Neurochem. 2001;79:152–160
11. Kuhn DM, Aretha CW, Geddes TJ. Peroxynitrite inactivation of tyrosine hydroxylase: mediation by sulfhydryl oxidation, not tyrosine nitration. The Journal of neuroscience : the official journal of the Society for Neuroscience. 1999;19:10289–10294
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15. Fukami G, Hashimoto K, Koike K, Okamura N, Shimizu E, Iyo M. Effect of antioxidant N-acetyl-L-cysteine on behavioral changes and neurotoxicity in rats after administration of methamphetamine. Brain research. 2004;1016:90–95
16. Pontieri FE, Crane AM, Seiden LS, Kleven MS, Porrino LJ. Metabolic mapping of the effects of intravenous methamphetamine administration in freely moving rats. Psychopharmacology. 1990;102:175–182
17. Burrows KB, Gudelsky G, Yamamoto BK. Rapid and transient inhibition of mitochondrial function following methamphetamine or 3,4-methylenedioxymethamphetamine administration. European journal of pharmacology. 2000;398:11–18
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21. Yiangou Y, Facer P, Durrenberger P, Chessell IP, Naylor A, Bountra C, Banati RR, Anand P. COX-2, CB2 and P2X7-immunoreactivities are increased in activated microglial cells/macrophages of multiple sclerosis and amyotrophic lateral sclerosis spinal cord. BMC neurology. 2006;6:12