1. We start with a non-abelian gauge theory. The covariant derivative is
$$D~=~\mathrm{d}+A, \qquad A~=~\mathrm{d}x^{\mu} A_{\mu},\tag{A}$$ 
while the field strength is 
$$\begin{align} \frac{1}{2}F_{\mu\nu}\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu} ~=~&F~=~D \wedge D\cr
~=~&\frac{1}{2}[D\stackrel{\wedge}{,}D]\cr
~=~&[\mathrm{d},A] + \frac{1}{2}[A\stackrel{\wedge}{,}A]\cr
~=~&\mathrm{d}A + A \wedge A, \end{align} \tag{6.35}$$
$$ F_{\mu\nu}~=~\partial_{[\mu}A_{\nu]} + [A_{\mu},A_{\nu}]. \tag{6.36}$$

2. Next consider a non-abelian Wilson-line$^1$ 
$$ U(t_2,t_1)~=~ \left\{\begin{array}{rcl} T\exp\left(-\int_{t_1}^{t_2}\! A\right)&{\rm for}& t_1\leq t_2,\cr AT\exp\left(-\int_{t_1}^{t_2}\! A\right)&{\rm for}& t_2\leq t_1,\end{array}\right.  \tag{7.1'} $$
over a (possibly open) curve $C$. Here $(A)T$ denotes [(anti)time-ordering](https://en.wikipedia.org/wiki/Ordered_exponential). Let us for simplicity assume from now on that $t_1\leq t_2$. Then we may write
$$ U(C)~=~T\exp\left(-\int_C\! A\right) \tag{7.1'} $$
with a parametrized curve $C:[t_1,t_2]\to\mathbb{R}^4$.

   The Wilson-line (7.1') is the solution to the following ODE
$$\begin{align} 
\frac{dU(t_2,t_1)}{dt_2}
~=~&-\dot{x}^{\mu}(t_2) A_{\mu}(t_2) U(t_2,t_1), \cr 
\frac{dU(t_2,t_1)}{dt_1}
~=~&U(t_2,t_1)\dot{x}^{\mu}(t_1) A_{\mu}(t_1),\cr  
U(t_1,t_1)~=~&{\bf 1}.\end{align}\tag{B}$$

3. We now make an infinitesimal variation of the curve $C$ to a new curve $C^{\prime}$. The varied curve $C^{\prime}$ is assumed to have the same end points as $C$, and the same parametrization interval $[t_1,t_2]$. We may define an infinitesimally thin 2-surface $\Sigma$ with oriented boundary 
$$ \partial \Sigma~=~ C^{\prime}-C \tag{C}$$
given by the two curves $C$ and $C^{\prime}$. This induces a (passive) change $\delta A$ of the gauge field $A$.

   NB: Be aware that the 2 sides 
$$   \int_{C}\! \delta A ~=~ \int_{C^{\prime}}\! A-\int_{C} \!A  ~=~  \oint_{\partial\Sigma} \!A ~=~ \iint_{\Sigma}\! \mathrm{d} A \tag{D} $$
and
$$
\iint_{\Sigma}\! F ~=~ \int_C\! \delta x^{\mu} F_{\mu\nu}   \mathrm{d} x^{\nu}  \tag{E} $$
of [Stokes' circulation theorem](https://en.wikipedia.org/wiki/Stokes%27_theorem) are _not_ necessarily equal for non-Abelian gauge-fields.$^2$ 

4. The infinitesimal (passive) change in holonomy is
$$\begin{align}  \delta U(C)~=~~~~&U(C^{\prime})-U(C)\cr
~\stackrel{(7.1')}{=}~~&-T\left[\exp\left(-\int_C\! A\right)\int_C\! \delta A\right]\cr
~\stackrel{(7.1')}{=}~~&-\int_{t_1}^{t_2}\! dt~U(t_2,t)\delta[\dot{x}^{\mu}(t)A_{\mu}(t)]U(t,t_1)\cr
~=~~~~&-\int_{t_1}^{t_2}\! dt~U(t_2,t)\left[\frac{d\delta x^{\mu}(t)}{dt}A_{\mu}(t)+\dot{x}^{\mu}(t)\delta A_{\mu}(t)\right]U(t,t_1)\cr
~\stackrel{\text{IBP}}{=}~~~&
\text{bulk terms} ~+~ \text{boundary terms},\end{align}\tag{F}$$
where
$$\begin{align}
\text{bulk}&\text{ terms}\cr
~=~&\int_{t_1}^{t_2}\! dt~U(t_2,t)\left[
\frac{\stackrel{\leftarrow}{d}}{dt}\delta x^{\mu}(t)A_{\mu}(t)
+\delta x^{\mu}(t)\dot{A}_{\mu}(t)\right.\cr
&\left. -\dot{x}^{\mu}(t)\delta A_{\mu}(t)
+\delta x^{\mu}(t)A_{\mu}(t)\frac{\stackrel{\rightarrow}{d}}{dt}
\right]U(t,t_1)\cr
~\stackrel{(B)}{=}~&\int_{t_1}^{t_2}\! dt~U(t_2,t)\left[
\dot{x}^{\nu}(t) A_{\nu}(t)\delta x^{\mu}(t)A_{\mu}(t)
+\delta x^{\mu}(t)\dot{x}^{\nu}(t)\partial_{\nu}A_{\mu}(t)\right.\cr
&\left. -\dot{x}^{\mu}(t)\delta x^{\nu}(t)\partial_{\nu} A_{\mu}(t)
-\delta x^{\mu}(t)A_{\mu}(t)\dot{x}^{\nu}(t) A_{\nu}(t)
\right]U(t,t_1)\cr
~\stackrel{(6.36)}{=}&\int_{t_1}^{t_2}\! dt~U(t_2,t)
\dot{x}^{\mu}(t) F_{\mu\nu}(t)\delta x^{\nu}(t)
U(t,t_1)\cr
~=~&\int_{t_1}^{t_2}\! dt~
\dot{x}^{\mu}(t) \underbrace{U(t_2,t)F_{\mu\nu}(t)U(t,t_1)}_{=:{\cal F}_{\mu\nu}(t)}\delta x^{\nu}(t)
\cr
~=~&T\left[\exp\left(-\int_C\! A\right)
\int_C\! F_{\mu\nu}\mathrm{d}x^{\mu} \delta x^{\nu}\right]
\cr
~\stackrel{(E)}{=}~&-T\left[\exp\left(-\int_C\! A\right)
\iint_{\Sigma}\! F\right]
,\end{align}\tag{7.25'}$$  
and
$$\begin{align}
\text{boundary terms}~=~&-\left[U(t_2,t)\delta x^{\mu}(t)A_{\mu}(t)U(t,t_1)\right]_{t=t_1}^{t=t_2}\cr
~=~& U(t_2,t_1)\delta x^{\mu}(t_1)A_{\mu}(t_1)
-\delta x^{\mu}(t_2)A_{\mu}(t_2)U(t_2,t_1)\cr
~\stackrel{(H)}{=}~&0,\end{align}\tag{G}$$
since the endpoints are not varied
$$ \delta x^{\mu}(t_1)~=~0~=~\delta x^{\mu}(t_2).\tag{H}$$
Eq. (7.25') answers OP's main question about eq. (7.25). The minus signs are caused by different sign conventions, such as choice of orientation.  

References:

1. A.M. Polyakov, _Gauge Fields and Strings,_ 1987; Chapter 7.


--

$^1$ A Wilson line is physics jargon for [holonomy](https://en.wikipedia.org/wiki/Holonomy). If the curve $C$ is closed, we speak of a Wilson loop rather than a Wilson line. We prefer to use time-ordering rather than path-ordering, since the latter is ambiguous. Ref. 1. uses a [path-ordering](https://en.wikipedia.org/wiki/Path-ordering) $P$ from left to right,
 
$$ \Psi(C)~:=~ P e^{\int_{C} \!A},  \tag{7.1} $$

which induces an opposite sign in front of the gauge field $A$ as compared to eq. (7.1').

$^2$ Let us mention for completeness that there exists a
[non-Abelian Stokes' Theorem](https://ncatlab.org/nlab/show/nonabelian+Stokes+theorem), which takes an
exponentiated form 

$$  Te^{-\oint_{\partial\Sigma} \! A}~=~ P_2\prod_{x\in\Sigma} U(\ast,x) e^{\iint_{\delta\Sigma_x}\!F}U(x,\ast)
~=:~P_2e^{\iint_{\Sigma}\!{\cal F}}. \tag{I}$$
Here all infinitesimal plaquettes $\delta\Sigma_x$ at the point $x$ are parallel-transported to the same fiducial base point $\ast$. The construction depends on a choice of surface ordering $P_2$. Recall that for an infinitesimal plaquette $\Sigma$, $$Te^{-\oint_{\partial\Sigma} \! A}  ~=~e^{\iint_{\Sigma}\!F},\tag{J}$$
cf .e.g. my Phys.SE answer [here](https://physics.stackexchange.com/a/688833/2451).