The Dalitz plot analysis technique is used to study the resonant substructures of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{D}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays in a data sample corresponding to $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $pp$ collision data recorded by the LHCb experiment during 2011 and 2012. A model-independent analysis of the angular moments demonstrates the presence of resonances with spins 1, 2 and 3 at high ${D}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ mass. The data are fitted with an amplitude model composed of a quasi-model-independent function to describe the ${D}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ S wave together with virtual contributions from the ${D}^{*}(2007{)}^{0}$ and ${B}^{*0}$ states, and components corresponding to the ${D}_{2}^{*}(2460{)}^{0}$, ${D}_{1}^{*}(2680{)}^{0}$, ${D}_{3}^{*}(2760{)}^{0}$ and ${D}_{2}^{*}(3000{)}^{0}$ resonances. The masses and widths of these resonances are determined together with the branching fractions for their production in ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{D}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays. The ${D}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ S wave has phase motion consistent with that expected due to the presence of the ${D}_{0}^{*}(2400{)}^{0}$ state. These results constitute the first observations of the ${D}_{3}^{*}(2760{)}^{0}$ and ${D}_{2}^{*}(3000{)}^{0}$ resonances, with significances of $10\ensuremath{\sigma}$ and $6.6\ensuremath{\sigma}$, respectively.