The main reason for the peak in the reactivities is a nuclear resonance effect. To show this, it's worth looking into the rich physics of cross-sections (see <PERSON> & <PERSON>, Nuclear Fusion, 1992). There are three main factors that affect fusion cross-sections (plotted above with data from Bosch & Hale), \begin{equation} \sigma = S(E) \frac{1}{E}\exp{\left(-\frac{B_G}{\sqrt{E}}\right)}\,. \end{equation}
1. The factor $\frac{1}{E}$ derives from the effective size of a particle due to its de Broglie wavelength, \begin{equation} \text{effective size} \sim \pi \lambda_{\mathrm{dB}}^2 = \pi\frac{h^2}{p^2} =\pi \frac{h^2}{2mE} \propto \frac{1}{E}\,. \end{equation}
2. The exponential factor derives from the tunneling probability through the potential barrier created by the Coulomb repulsion between the reactants, \begin{equation} \text{tunneling probability} \propto \exp{\left(-\frac{\pi \alpha Z_1 Z_2 \sqrt{2m_rc^2}}{\sqrt{E}}\right)} \equiv \exp{\left(-\frac{B_G}{\sqrt{E}}\right)}\,, \end{equation} where $Z_1$ and $Z_2$ are the atomic numbers of the particles, $m_r$ is the reduced mass of the system, $\alpha={k_e e^2}/{\hbar c}$ is the fine structure constant, and $B_G$ is known as the Gamow constant (well-explained in this wiki entry).
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The factor $S(E)$ is the so-called S-function and was introduced by astrophysicists to capture the remaining, relatively slowly varying nuclear physics contribution to the cross-section. In fact, the S-function varies so slowly that it can be plotted on linear scales (plotted below with data from Bosch & Hale). Hence it is often more useful to compare the S-functions of similar reactions, rather than the cross sections. (For more information about the astrophysical origin of the S-function see this review article by <PERSON> et al., Rev. Mod. Phys., 1957.)
The peaks in the S-functions are due to resonances, which arise only at certain energies when the relative phase and amplitude of the internal bound wavefunction and external traveling wavefunction of the quasi-particle match well and facilitate tunneling. This causes the cross sections and reactivities to peak for DT and $\mathrm{D}\,^3\mathrm{He}$, while the DD reaction is far from resonance in the plotted energy range. 469