A real flag manifold is $M=G/P$ where $G$ is a real semisimple Lie group and $P$ is a parabolic subgroup of $G$. Here parabolic means that the Lie algebra
$\mathfrak p$ contains a minimal parabolic subalgebra $\mathfrak p_{min}$ of $\mathfrak g$.
In order to define this, let $\mathfrak g=\mathfrak k+\mathfrak s$ be a Cartan decomposition, $\mathfrak a$ is a maximal
Abelian subspace of $\mathfrak s$,
$\mathfrak g =\mathfrak g_0+\sum_{\lambda\in\Sigma} \mathfrak g_\lambda$ be the root decomposition with respect to $\mathfrak a$.
We have $\mathfrak g_0=\mathfrak m+\mathfrak a$ where $\mathfrak m$ is the
centralizer of $\mathfrak a$ in $\mathfrak k$. Choose a positive subsystem
$\Sigma^+$ and let $\mathfrak n = \sum_{\lambda\in\Sigma^+}\mathfrak g_\lambda$
(nilpotent Lie algebra). We get the Iwasawa decomposition
$\mathfrak g =\mathfrak k+\mathfrak a+\mathfrak n$. Finally,
$\mathfrak p_{min}=\mathfrak m+\mathfrak a+\mathfrak n$.
In general, $\mathfrak p = \mathfrak p_{min} + \sum_{\lambda\in\langle\Phi\rangle}\mathfrak g_{-\lambda}$, where $\Phi$ is a subset of of the system of simple roots of $\Sigma$.
It turns out $K$, as a subgroup of $G$, acts by left translations on $G/P$
and this action is transitive. In fact, the isotropy algebra at the basepoint
is $\mathfrak k\cap\mathfrak p$, and if you count dimensions, you see that the
orbit is open. It is also closed by compactness of $K$, so $K/K\cap P=G/P$.
It remains to identify the left hand side with an orbit of the isotropy representation of the symmetric space $G/K$, namely, the action of $K$ on $\mathfrak s$, but this is not hard.
I can refer you to the nice book "An Introduction to Lie Groups and the Geometry of Homogeneous Spaces", by Andreas Arvanitoyeorgos (http://www.ams.org/bookstore?fn=20&arg1=stmlseries&ikey=STML-22).
Edit: Indeed $\mathfrak k\cap\mathfrak p =\mathfrak m + \sum_{\lambda\in\langle\Phi\rangle}(\mathfrak g_{\lambda}+\mathfrak g_{-\lambda})\cap\mathfrak k$ and you can select a point $x\in\mathfrak a\subset\mathfrak s$
such that $\lambda(x)=0$ if and only if $\lambda\in\langle\Phi\rangle$.
The desired $K$-orbit is that through $x$.