= Solution
First the tertiary amine attacks the primary allylic bromide carbon by <SN2 reaction>, giving the quaternary ammonium salt. The less hindered nitrogen face is favored by the pre-existing fused-ring stereochemistry.
DBU removes the proton at the carbon alpha to both the ammonium nitrogen and the <ester> <carbonyl>. The resulting ammonium <ylide> undergoes a concerted <2,3-sigmatropic rearrangement>: the N–CH$_2$(allyl) bond breaks, a bond forms between the <ylide> carbon and the \b[aryl-bearing terminal allyl carbon], and the double bond moves to give $\mathrm{-CH(Ar)-C(Me)=CH_2}$. Nitrogen becomes neutral again. An ordinary alkyl substitution would attach the wrong end of the allyl unit.
An envelope-like five-membered transition structure keeps Ar outside the crowded bicyclic framework. With the pre-existing OMe dash and ring-junction geometry fixed, place the newly installed side chain on the solid-wedge face at the former <ylide> carbon and choose the benzylic configuration that places Ar outward, as in the displayed intermediate. This is diastereoselective, suprafacial allyl transfer.
Finally methanesulfonic acid protonates the methyl <acetal>. Methanol leaves to give an oxocarbenium/iminium-stabilized electrophilic centre, which is attacked intramolecularly by the electron-rich arene at its available ortho position. <Deprotonation> restores aromaticity and closes the bridge. The terminal isopropenyl group survives; it is \b[not] the bond consumed in this final <Intramolecular Friedel-Crafts reaction>. The constrained tether selects the displayed bridged diastereomer.
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