I refer you to this set of presentation slides. On slide 8, the author explains the preference for the staggered conformation of ethane by saying that there is a favourable hyperconjugative interaction between $\sigma^{}_{\ce{C-H}}$ of the $\ce{C-H}$ bond in one methyl group and $\sigma^{*}_{\ce{C-H}}$ of another $\ce{C-H}$ bond in the other methyl group.
However, I am puzzled by the diagram. Since both $\ce{C-H}$ bonds are similar in terms of bonding, their bonding and antibonding $\sigma$ molecular orbitals should have the same energies. Instead, the diagram shows that one of the $\sigma_{\ce{C-H}}$ bonds is lower than the other in energy. How can this be right? I have shown a screenshot of the diagram below.
Also, how can a $\sigma_{\ce{C-H}}$ donate electron density to a $\sigma^*_{\ce{C-H}}$ effectively? After all, the energy difference between them is rather significant since the orbital interaction between the atomic orbitals of $\ce{C}$ and $\ce{H}$ is large.
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