How do I perform the sifting process in empirical mode decomposition?
It's not shifting, it's sifting. After you find the $m_1$ curve, your subtract it from your signal $X$ to get $h_1= X(t) - m_1$. You're essentially taking away that signal component and are left with $h_1$, which is your new starting point of your next iteration. I assume you were confused by the terminology.

Sifting is a general term in signal processing related to separating out components of a signal one at a time. It is frequently used in context of wavelet decomposition since this process is very similar in this respect. 

The reason the next function is $h_{11}$ and not $h_2$ is because you have to keep going with this process until the resulting $h_{1k}$ is an IMF function (criteria given on the top of the page). Once you've reached that point, you get your first IMF basis function $c_1$. Only at this point have you reached your first empirical mode.

Then you start over by finding $m_2$ and so on.