One obtains those expressions by considering a particular action of the Poincare group on fields.

Consider, for example, a single real scalar field $\phi:\mathbb R^{3,1}\to\mathbb R$.  Let $\mathcal F$ denote the space of such fields.  Define an action $\rho_\mathcal F$ of $P(3,1)$ acting on $\mathcal F$ as follows
\begin{align}
  \rho_\mathcal F(\Lambda,a)(\phi)(x) = \phi(\Lambda^{-1} (x-a))
\end{align}
Sometimes people will write this as $\phi'(x) = \phi(\Lambda^{-1} x)$ for brevity.  Now let $G$ denote a generator of the Lie algebra of the Poincare group (namely an element of a chosen basis for this Lie algebra).  We can use this generator to define a corresponding infinitesimal generator for group action $\rho_\mathcal F$ as follows:
\begin{align}
  G_\mathcal F(\phi) = i\frac{\partial}{\partial\epsilon}\rho_\mathcal F(e^{-i\epsilon G})(\phi)\bigg|_{\epsilon = 0}
\end{align}

**Example - translations.** Consider the translation generators $P^\mu$ which have the property
\begin{align}
  e^{-ia_\mu P^\mu}x = x+a
\end{align}
The generator of $\rho_\mathcal F$ corresponding to $P^0$, for instance, is
\begin{align}
  (P^0)_\mathcal F(\phi)(x) 
&= i\frac{\partial}{\partial\epsilon}\rho_\mathcal F(e^{-i\epsilon P^0})(\phi)\bigg|_{\epsilon = 0} \\
&= i\frac{\partial}{\partial\epsilon}\phi(x + \epsilon e_0)\bigg|_{\epsilon = 0} \\
&= i\partial^0\phi(x)
\end{align}
where $e_0 = (1,0,0,0)$, and similarly for the other $P^\mu$, which gives
\begin{align}
  (P^\mu)_\mathcal F = i\partial^\mu.
\end{align}

**Example - Lorentz boosts.**

If you use this same procedure for Lorentz boost generators, you will find that
\begin{align}
  (M^{\mu\nu})_\mathcal F = i(x^\mu\partial^\nu - x^\nu\partial^\mu) = x^\mu p^\nu - x^\nu p^\mu
\end{align}

**Disclaimer about signs etc.** There are a lot of conventional factors of $i$ and negative signs floating around which I wasn't super careful to keep track of, if you notice an error in this regard, please let me know and I'll fix it.