Mechanism of arene side chain oxidation by permanganate
Some general information on side-chain oxidation in alkylbenzenes is available at Chemguide:


  An alkylbenzene is simply a benzene ring with an alkyl group attached
  to it. Methylbenzene is the simplest alkylbenzene.
  
  Alkyl groups are usually fairly resistant to oxidation. However, when they are attached to a benzene ring, they are easily oxidised by
  an alkaline solution of potassium manganate(VII) (potassium
  permanganate).
  
  Methylbenzene is heated under reflux with a solution of potassium
  manganate(VII) made alkaline with sodium carbonate. The purple colour
  of the potassium manganate(VII) is eventually replaced by a dark brown
  precipitate of manganese(IV) oxide.
  
  The mixture is finally acidified with dilute sulfuric acid.
  
  Overall, the methylbenzene is oxidised to benzoic acid.
  
  
  
  Interestingly, any alkyl group is oxidised back to a -COOH group on
  the ring under these conditions. So, for example, propylbenzene is
  also oxidised to benzoic acid.
  
  


Regarding the mechanism, a Ph.D. student at the University of British Columbia did his doctorate on the mechanisms of permanganate oxidation of various organic substrates.1 Quoting from the abstract:


  It was found that the most vigorous oxidant was the permanganyl ion ($\ce{MnO3+}$), with some contributing oxidation by both permanganic acid ($\ce{HMnO4}$) and permanganate ion ($\ce{MnO4-}$) in the case of easily oxidized compounds such as alcohols, aldehydes, or enols.


The oxidation of toluene to benzoic acid was one of the reactions investigated, and a proposed reaction mechanism (on pp 137–8) was as follows. In the slow step, the active oxidant $\ce{MnO3+}$ abstracts a benzylic hydrogen from the organic substrate.


  $$\begin{align}
\ce{2H+ + MnO4- &amp;&lt;=&gt; MnO3+ + H2O} &amp; &amp;\text{(fast)} \\
\ce{MnO3+ + PhCR2H &amp;-&gt; [PhCR2^. + HMnO3+] &amp; &amp;\text{(slow)}} \\
\ce{[PhCR2^. + HMnO3+] &amp;-&gt; PhCR2OH + Mn^V} &amp; &amp;\text{(fast)} \\
\ce{PhCR2OH + Mn^{VII} &amp;-&gt; aldehyde or ketone} &amp; &amp;\text{(fast)} \\
\ce{aldehyde + Mn^{VII} &amp;-&gt; benzoic acid} &amp; &amp;\text{(fast)} \\
\ce{ketone + Mn^{VII} &amp;-&gt; benzoic acid} &amp; &amp;\text{(slow)} \\
\ce{5 Mn^V &amp;-&gt; 2Mn^{II} + 3Mn^{VII}} &amp; &amp;\text{(fast)}
\end{align}$$


The abstraction of a benzylic hydrogen atom is consistent with the fact that arenes with no benzylic hydrogens, such as tert-butylbenzene, do not get oxidised.



Reference


Spitzer, U. A. The Mechanism of Permanganate Oxidation of Alkanes, Arenes and Related Compounds. Ph.D. Thesis, The University of British Columbia, November 1972. DOI: 10.14288/1.0060242.