$\newcommand{\bl}[1]{\boldsymbol{#1}}
\newcommand{\e}{\bl=}
\newcommand{\p}{\bl+}
\newcommand{\m}{\bl-}
\newcommand{\mb}[1]{\mathbf {#1}}
\newcommand{\mr}[1]{\mathrm {#1}}
\newcommand{\plr}[1]{\left(#1\right)}
\newcommand{\blr}[1]{\left[#1\right]}
\newcommand{\clr}[1]{\left\{#1\right\}}
\newcommand{\vlr}[1]{\left\vert#1\right\vert}
\newcommand{\vra}[1]{\left|#1\right\rangle}
\newcommand{\lavra}[2]{\left\langle#1|#2\right\rangle}
\newcommand{\vp}{\vphantom{\dfrac{a}{b}}}
\newcommand{\Vp}[1]{\vphantom{#1}}
\newcommand{\tl}[1]{\tag{#1}\label{#1}}$
Although mathematical and not physical I like the interpretation of inner product given by Feynman in his lectures on Quantum Mechanics.
Consider that the Hilbert space of states is of finite dimension $\:n\:$ and let a complete orthonormal basis of states $\:\clr{\vra{\mb e_1},\vra{\mb e_2},\cdots,\vra{\mb e_k},\cdots,\vra{\mb e_n}}$. For normalized quantum states $\:\vra{\bl \psi_\mr A}\:$ and $\:\vra{\bl \psi_\mr B}\:$ we have
\begin{align}
\!\!\!\!\vra{\bl \psi_\mr A} & \e \mr a_1\vra{\mb e_1} \p \mr a_2 \vra{\mb e_2} \p \cdots \mr a_k \vra{\mb e_k}\p\cdots\mr a_n\vra{\mb e_n}\,, \:\:\sum\limits_\imath \vlr{\mr a_\imath}^2 \e 1\,, \:\: \mr a_\imath \bl\in \mathbb C
\tl{01.a}\\
\!\!\!\!\vra{\bl \psi_\mr B} & \e \mr b_1\vra{\mb e_1} \p \mr b_2 \vra{\mb e_2} \p \cdots \mr b_k \vra{\mb e_k}\p\cdots\mr b_n\vra{\mb e_n}\,, \:\: \sum\limits_\jmath \vlr{\mr b_\jmath}^2 \e 1\,, \:\: \mr b_\jmath \bl\in \mathbb C
\tl{01.b}
\end{align}
For the probability to find the particle in state $\:\vra{\bl \psi_\mr A}\:$ if it starts in state $\:\vra{\bl \psi_\mr B}\:$ we use probability amplitudes and the laws of combining them, see Figure-01.
The probability amplitude for the transition from the state $\:\vra{\bl \psi_\mr B}\:$ to the basic state $\:\vra{\mb e_1}\:$ is $\:\mr b_1$, while in turn the probability amplitude for the transition from the basic state $\:\vra{\mb e_1}\:$ to the state $\:\vra{\bl \psi_\mr A}\:$ is $\:\mr a^{\bl *}_1$,$\:\:$the complex conjugate of $\:\mr a_1$. The amplitude for the transition from the state $\:\vra{\bl \psi_\mr B}\:$ to state $\:\vra{\bl \psi_\mr A}\:$ passing through the intermediate basic state $\:\vra{\mb e_1}\:$ is the product of the amplitudes $\:\mr b_1\,\mr a^{\bl *}_1$. So the amplitude for the transition from the state $\:\vra{\bl \psi_\mr B}\:$ to state $\:\vra{\bl \psi_\mr A}\:$ passing through any intermediate basic state $\:\vra{\mb e_k}\:$ is $\:\mr b_k\,\mr a^{\bl *}_k$.
\begin{equation}
\vra{\bl \psi_\mr B}\stackrel{b_k}{\bl{\m\!\!\!\m\!\!\!\m\!\!\!\longrightarrow}}\vra{\mb e_k}\stackrel{\mr a^{\bl *}_k}{\bl{\m\!\!\!\m\!\!\!\m\!\!\!\longrightarrow}}\vra{\bl \psi_\mr A}
\tl{02}
\end{equation}
Since passing through any basic state $\:\vra{\mb e_k}\,\plr{k\e 1,2,\cdots,n}\:$ is a complete set of independent possibilities, the probability amplitude for the transition from the state $\:\vra{\bl \psi_\mr B}\:$ to the basic state $\:\vra{\bl \psi_\mr A}\:$ is the sum of the amplitudes $\:\mr b_k\,\mr a^{\bl *}_k$
\begin{equation}
\mr b_1\,\mr a^{\bl *}_1\p\mr b_2\,\mr a^{\bl *}_2\cdots\p\mr b_n\,\mr a^{\bl *}_n \e\sum\limits_{k\e 1}^{k\e n}\mr b_k\,\mr a^{\bl *}_k\e
\begin{bmatrix}
\mr a^{\bl *}_1 & \mr a^{\bl *}_2 & \cdots & \mr a^{\bl *}_n\Vp{\tfrac{a}{b}}
\end{bmatrix}
\begin{bmatrix}
\mr b_1 \Vp{\tfrac{a}{b}}\\
\mr b_2 \Vp{\tfrac{a}{b}}\\
\vdots \Vp{\tfrac{a}{b}}\\
\mr b_n \Vp{\tfrac{a}{b}}
\end{bmatrix}\e \lavra{\bl \psi_\mr A}{\bl \psi_\mr B}
\tl{03}
\end{equation}
Equation \eqref{03} is given schematically in Figure-02.
We know that the inner product $\:\lavra{\bl \psi_\mr A}{\bl \psi_\mr B}\:$ is invariant with respect to the choice of the complete basis $\:\clr{\vra{\mb e_1},\vra{\mb e_2},\cdots,\vra{\mb e_k},\cdots,\vra{\mb e_n}}$.
The probability for the transition $\:\vra{\bl \psi_\mr B}\bl{\longrightarrow}\vra{\bl \psi_\mr A}\:$ is
\begin{equation}
\mr P\plr{\vra{\bl \psi_\mr B}\bl{\longrightarrow}\vra{\bl \psi_\mr A}\vp} \e \vlr{\lavra{\bl \psi_\mr A}{\bl \psi_\mr B}\vp}^2
\tl{04}
\end{equation}
