Roughly and intuitively speaking, a self financing strategy is (for me) a trading strategy which requires no extra cost during the trading except for the initial capital. Suppose that we are in a continuous time case. Let $(\Omega,\mathcal{F},P)$ be a prob space with filtration $\mathbb{F}$, $S$ (a semimartingle) be the discounted price process of the stock, $B$ the bank account ($B \equiv 1$), $\phi = (\theta, \eta)$ be a trading strategy, where $\eta$ is the number of units held in the bank account and $\theta$ (predictable process) in the stock $S$. Let $V$ be the value of the portfolio. The total cost of the strategy $\phi$ on $[0,t]$ is defined as
\begin{equation}
C_t(\phi):= V_t(\phi) -(\theta\bullet S)_t,
\end{equation}
where $(\theta\bullet S)_t$ (stochastic integral) denote the gain up to time $t$. In the lecture of mathematical finance that I followed, we defined $\phi$ to be self-financing if for all $t\ge 0$
\begin{equation}
C_{t}(\phi)=C_0(\phi) \, P\text{-a.s.}.
\end{equation}
Note that $C_0(\phi) := V_0$ is the initial capital. Now it is possible to prove the following.
Lemma $\phi$ is self-financing iff
\begin{equation}
V(\phi)=V_0(\phi) + (\theta\bullet S).
\end{equation}
Indeed, there exists a bijection between self-financing strategies $\phi = (\theta,\eta)$ and the pairs $(V_0,\theta)$, where $V_0 ∈ L_0(\mathcal{F}_0)$ and $\theta$ is predictable and $S$-integrable. Explicitly, $V_0 = V_0(φ)$ and
\begin{equation}
\eta = V_0 + (\theta\bullet S) − \theta \cdot S,
\end{equation}
where $\theta \cdot S$ denotes a scalar product. Moreover, if $\phi = (\theta, \eta)$ is self-financing, then $\eta$ is also predictable.
The differential form of $V(\phi)=V_0(\phi) + (\theta\bullet S)$ is your definition of self financing strategy. I hope that it may be useful.
As requested in your comment, I add some details. It is not necessary to work with a discounted price process, but it make our life easier and it simplify the computations (and there is no loss of generality). Moreover, in order to understand the intuition behind the concept of self-financing strategy it is not relevant if we consider discounted or undiscounted price process. (In my opinion, in order to understand the intuition behind the concepts of mathematical finance it may be useful to study the discrete time case).
To say that a portfolio is self-financing means that when we rearrange the portfolio at time $t$ (e.g. from $\phi_{t-1}$ to $\phi_{t}$ in a discrete time model) there is no input/outflow of money. This means that you can rearrange your strategy using just the money which comes from the initial capital and the gain up to time $t$.
In order to motivate the above definition of self-financing strategy, we can look at the following (intuitive) argument. Suppose we keep a strategy $\phi=(\theta, \eta)$ constant between $t$ and $t+\Delta t$ and only change it from $\phi_t$ to $\phi_{t+\Delta t}$ at time $t$. Then in the interval $(t,\Delta t]$ the cost of this trading strategy is given by
\begin{align}
C_{t+\Delta t}-C_{t}&=(\phi_{t+\Delta t}-\phi_{t})\cdot(S_t,1) \\
&=\theta_{t+\Delta t}S_t +\eta_{t+\Delta t} -\theta_{t}S_t -\eta_t \\
&=\theta_{t+\Delta t}S_t +\eta_{t+\Delta t} -\theta_{t}S_t -\eta_t + [-\theta_{t+\Delta t}S_{t+\Delta t}+\theta_{t+\Delta t}S_{t+\Delta t}] \\
&=V_{t+\Delta t}-V_{t}-\theta_{t+\Delta t}(S_{t+\Delta t}-S_t),
\end{align}
recall that in discounted term the value of the portfolio is defined as $V_t := \theta_tS_t+\eta_t$. Summing up and taking $\Delta t$ small suggests the above (natural) definition of cumulative cost process,
\begin{equation}
C_t(\phi):= V_t(\phi) - \int_{0}^t\theta_udS_u.
\end{equation}
Hence, we call $\phi$ self-financing if $C_t(\phi)\equiv C_0(\phi)$ $P$-a.s. for all $t\ge0$, which means, according to our intuition, that the total cost of the strategy $\phi$ is known at time zero and we are sure that no extra money is required in the future.
In my opinion the same argumentation can be applied to the undiscounted case $\hat{B}\not\equiv 1$, which should give for the undiscounted pair $(\hat{S}_t,\hat{B}_t)$ the following definition of total cost
\begin{align}
\hat{C}_t := \hat{V}_t(\phi)-\int_{0}^t\theta_ud\hat{S}_u-\int_{0}^{t}\eta_ud\hat{B}_u.
\end{align}
Thus, we can conclude $\phi$ self-financing iff
\begin{equation}
\hat{V}_t(\phi)= \hat{V}_0 +\int_{0}^t\theta_ud\hat{S}_u +\int_{0}^{t}\eta_ud\hat{B}_u,
\end{equation}
whose differential form is your definition.
