Why the energies of orbitals in the same subshell decrease with increase of atomic number?
Short answer: more protons in the nucleus means higher nuclear charge which means stronger attraction between the electron and the nucleus.

Long answer: The energy of the orbital is driven by the Hamiltonian operator $\hat{H} = \hat{T} + \hat{V}$ where $\hat{V}$ is the potential energy. For a simple 1-electron atom/ion, the potential energy is the electrostatic attraction between the positively charged nucleus $Q$ and the negative electron $q$: $V = \frac{1}{4\pi\epsilon_0} \frac{Qq}{r}$. Lithium has 3 protons in the nucleus versus the 1 proton for hydrogen, so $Q$ is three times higher and the potential energy of the 1s orbital in $\ce{Li^{2+}}$ is roughly three times more negative than the 1s orbital in $\ce{H}$ (remember $q$ is negative).

If you have more than one electron, the potential energy becomes more complicated because you have to include the electrostatic repulsion from all the other electrons as well, but the general trend still holds, all else being equal.