I once had the same question as you and I did go to my chemistry teacher for an answer. This is his original response:
"In hydrogen, all orbitals with the same principal quantum number 'n' (1,2,3...) are degenerate, regardless of the orbital angular momentum quantum number 'l' (0,1...n-1 or s,p,d,). However, in atoms with more than one electron, orbitals with different values of l for a given value for n are not degenerate. Why is this? Surely the radial distribution functions are similar for hydrogen (in that there's still penetration of orbitals and so on). Or is it that orbitals with different values of l are degenerate for a value of n greater than would be occupied in that particular atom's ground state?
The answer is right there in your question. It's only the interaction of multiple electrons in an atom like He, Li, Be, etc. that makes the different angular momenta wave functions differ in energy.
Consider this. For the one electron system, why should a p or d orbital differ in energy from an s? What makes them differ?
In the multi-electron case, the p orbitals have different spatial extent, different angular components, so the electron density caused by an electron in that orbital will interact differently with the other electrons. In other words, you need to have more than one electron for the "shape" of the p and d and f orbitals to matter to the other electrons.
In the H atom, there's only one electron, so there's no electron-electron repulsion to differentiate the s, p, and d orbitals."
Hope this clarifies your doubt.