Before proceeding with the dual dynamics itself I would like to mention very good pedagogical reviews of the Double Field Theory appeared recently [1,2].
The so-called Duff's procedure implies a non-trivial feature of T-duality transformations: the hidden symmetry between equations of motion and Bianchi identities. It was suggested to understand left-moving and right-moving coordinates of a closed string as independent [1,2, for review of T-duality see 3]. And to collect them to a set of ordinary and dual coordinates. The so-called dual Lagrangian governs the dual dynamics. The key point is, equations of motion for the dual coordinates appear to be equivalent to Bianchi identities for the ordinary coordinates and vice versa. A relation between these coordinates leads to the notion of the generalised metric.
Equations of motion for the field $X^\mu$ that follow from the Polyakov action have the form of the conservation law $\partial_a\mathcal{G}^a_\mu=0$ for some current
\begin{equation}
\mathcal{G}^a_\mu=\left(\sqrt{-h}h^{ab}{G}_{\mu \nu}+\epsilon^{ab}B_{\mu\nu}\right)\partial_aX^\nu.
\end{equation}
Locally solutions of this equation can be represented by the Hodge dual of the full derivative $\mathcal{G}^a_\mu:=\epsilon^{ab}\partial_bY_\mu$ of the would-be dual coordinates $Y_\mu$.
Hence, taking the derivative $\partial_a$ of this expression one obtains the equations of motion for $X^\mu$ on the left hand side and the Bianchi identities $\epsilon^{ab}\partial_a\partial_bY_\mu=0$ for the field $Y_\mu$ on the right hand side.
Equations of motion for the field $X^\mu$ can be derived from the first order Lagrangian by introducing the extra independent field $U_a^\mu$
\begin{equation}
\mathcal{L}_x=\frac12(\sqrt{h}hG+\epsilon B)UU-(\sqrt{h}G+\epsilon B)U\partial X.
\end{equation}
Equations of motion for the fields $U_a^\mu$ and $X^\mu$ that follow from this Lagrangian give an algebraic constraint and a dynamical equation
\begin{equation}
\begin{aligned}
\frac{\partial \mathcal{L}_x}{\partial U_a^\mu}=0 & \Longrightarrow U_a^\mu-\partial_aX^\mu=0, \\
\partial_a \,\frac{\partial \mathcal{L}_x}{\partial\, \partial_a X^\mu}=0 & \Longrightarrow \partial_a\mathcal{G}^a_\mu=0.
\end{aligned}
\end{equation}
The next step is to introduce the dual Lagrangian for the field $Y_\mu$
\begin{equation}
\mathcal{L}_y=\frac12(\sqrt{h}hG+\epsilon B)UU-\epsilon^{ab}\partial_a Y_\mu U_b^\mu.
\end{equation}
Its variation with respect to $Y_\mu$ gives the Bianchi identities $\epsilon^{ab}\partial_a U_b^\mu\equiv \epsilon^{ab}\partial_a\partial_b X^\mu =0$ for the field $X^\mu$ while variation with respect to $U_a^\mu$ implies
\begin{equation}
\sqrt{-h}h^{ab}G_{\mu\nu}U_b^\nu +\epsilon^{ab}B_{\mu\nu}U_b^\nu=\epsilon^{ab}\partial_bY_\mu.
\end{equation}
Solving this equation with respect to $U_a^\mu$ we obtain
\begin{equation}
\label{EOMY}
\epsilon^{ab}U_b^\mu=\frac{1}{\sqrt{-h}}(h^{ab}P^{\mu \nu}+\epsilon^{ab}Q^{\mu \nu})\partial_b Y_\nu,
\end{equation}
where $P=(G-BG^{-1}B)^{-1}$ and $Q=(B-GB^{-1}G)^{-1}$. This expression has exactly the same form as EOM for $X^\mu$ but with $X^\mu$ replaced by $Y_\mu$ and $(G,B)$ by $(P,Q)$. Hence, the intermediate result is that doubling of the coordinates reveals hidden symmetry of equations of motion for a bosonic string and Bianchi identities.
To make this symmetry manifest it is useful to make the following definitions
\begin{equation}
\begin{aligned}
&\mathcal{G} =\sqrt{-h}h\cdot\partial Y& \mathcal{F}&=\sqrt{-h}h\cdot\partial X\\
&\tilde{\mathcal{G}}=G\mathcal{F}+B\tilde{\mathcal{F}} & \tilde{\mathcal{F}}&=\sqrt{-h}\epsilon\cdot\partial X .
\end{aligned}
\end{equation}
Then all equations that we dealt with are combined indices into one set introducing matrix notations
\begin{equation}
\label{EOM}
\eta_{MN}\tilde{\Phi}^{iN} =\mathcal{H}_{MN}\Phi^{iN},
\end{equation}
where the capital Latin indices $M,N=1\ldots 2n$. Here the objects $G$ and $F$ were collected into two $2n$-rows
\begin{eqnarray}
\tilde{\Phi}^{iM}=\begin{bmatrix}
\tilde{\mathcal{F}}\\
\tilde{\mathcal{G}}
\end{bmatrix}, & \Phi^{iM}= \begin{bmatrix}
\tilde{\mathcal{F}}\\
\tilde{\mathcal{G}}
\end{bmatrix}
\end{eqnarray}
and the $2n\times 2n$ matrices $\mathcal{H}$ and $\eta$ are schematically given by
\begin{eqnarray}
\mathcal{H}_{MN}=
\begin{bmatrix}
G^{ab} & GB \\
-GB & G_{mn}-BGB
\end{bmatrix}, &
\eta_{MN}=\begin{bmatrix}
0 &\delta\\
\delta & 0
\end{bmatrix}.
\end{eqnarray}
So, the idea is to introduce the so-called dual coordinates $Y$ that naturally complete the dynamics of string to an $SO(n,n)$ covariant form. In other words just to write equations in such a form as to see the hidden symmetry apparently.
The objects $\tilde{\Phi}$ and $\Phi$ transform in the fundamental representation of $SO(n,n)$, the matrix $\mathcal{H}$ transforms as a 2-tensor and the matrix $\eta_{MN}$ is an $SO(n,n)$ invariant tensor:
\begin{equation}
\label{Oddtransf}
\begin{aligned}
&\Phi'^{iM}=\mathcal{O}^M{}_N\Phi'^{iN}, & \mathcal{H}'_{MN} &= \mathcal{O}_M{}^K\mathcal{H}_{KL}O_N{}^L,\\
&\tilde{\Phi}'^{iM}=\mathcal{O}^M{}_N\tilde{\Phi}'^{iN},& \eta_{MN} &= \mathcal{O}_M{}^K\eta_{KL}O_N{}^L.
\end{aligned}
\end{equation}
Note that the last equation implies that $\mathcal{O}\in SO(n,n)$.
The matrix $\mathcal{H}_{MN}$ that is the so-called generalised metric allows to consider the ordinary metric $G$ and the 2-form field $B$ on an equal footing. Moreover, while T-duality is realised by non-linear transformations of the supergravity fields (Buscher rules) the generalised metric transforms linearly. One can check that the linear $SO(n,n)$ transformations of $\mathcal{H}_{MN}$ lead exactly to the Buscher rules.
There is nothing special of 1-dimensional objects, string, and the similar procedure can be repeated for $Dp$-branes or $M2$ and $M5$ branes. In the latter case one arrives to extended space for M-theory.
Obviously, this procedure doubles the number of degrees of freedom and one needs to introduce a constraint that returns us to $n$ coordinates. This constraint is known as section condition. One could think, that the constraint would be needed for construction of a supersymmetric version of the formalism to keep the balance between bosonic and fermionic d.o.f. However, there are papers [4,5], that demonstrate us how this can be done in duality invariant way.
