The structural phase transition in the $A15$ compounds are investigated theoretically with the standpoint that the band Jahn-Teller effect of the twofold-degenerate ${\ensuremath{\Gamma}}_{12}$ subbands crossing the Fermi level is responsible for the instability. On the basis of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ perturbation theory, the ${\ensuremath{\Gamma}}_{12}$ subbands are revealed to be well described by two parabolic bands which couple not only to the bulk distortions, but also to the displacements of the ${\ensuremath{\Gamma}}_{12}$ optic modes. It is found that when the electron-lattice coupling exceeds the threshold of strength, the tetragonal phase with almost the same stabilities of $\frac{c}{a>1}$ and $\frac{c}{a<1}$ appears, accompanying one of the ${\ensuremath{\Gamma}}_{12}$ optic modes below a weak first-order phase transition temperature ${T}_{M}$. The temperature dependences of the elastic moduli are calculated; it is found that ${c}_{11}\ensuremath{-}{c}_{12}$ vanishes below ${T}_{M}$ while ${c}_{33}\ensuremath{-}{c}_{13}$ recovers from its softening partially or completely with decreasing temperature below ${T}_{M}$. The long-wavelength acoustic phonons are also investigated in order to clarify the relation between the phonon anomalies and the structural transition. The [110]${\mathrm{T}}_{1}$ mode ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}\ensuremath{\parallel}[110]$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{e}}\ensuremath{\parallel}[1\overline{1}0]$) is considerably softened in the range $0<q\ensuremath{\lesssim}2{k}_{F}$. This softening begins at high temperatures, remaining even at absolute zero. The theory explains successfully the various aspects of the phase transitions in ${\mathrm{V}}_{3}$Si and ${\mathrm{Nb}}_{3}$Sn. The comparison between them proves that the second-order Jahn-Teller effect occurs in both compounds.