How does hyperconjugation explain stability of partial positive charges?
The solution is referring to the partial positive charge on the carbon opposite the developing C-B bond. It is stabilized by substituents similar to how a full carbocation is stabilized by substituents, e.g., in the tert-butyl carbocation. This doesn't need to be a completely empty p-orbital—electron density will be dynamically influenced by any imbalance in charge distribution—but if it's easier to conceptualize, you could imagine it as the pi-bond electrons being fully transferred to the C-B bond, leaving an empty p-orbital on the opposite carbon, which is stabilized by hyperconjugation.

The reality should be closer to a dynamic process at both sites, though. The pi-bond transfers electron density into the C-B bond, while the substituents stabilize the rising positive charge on the other carbon. As the C-B bond develops, the B-H bond simultaneously weakens, and begins moving its electron density toward the partial-positive carbon.