I've come across this relation numerous times, textbooks use it as if it is obvious. But I have never come across a proof or an intuitive explanation about why is it true.
It would be helpful if someone helps me with what exactly to refer to in order to understand this.
This follows from a fundamental posulate in quantum mechanics, that $\langle f \mid i \rangle$ is literally the probability amplitude that given the system is initially in the state $\mid i\rangle$, we find it in the state $\mid f\rangle$.
Since we are dealing with probability amplitudes, the probability for this process is then the square of this. That is, $$P_{i\rightarrow f}=\langle f \mid i \rangle ^2$$
Linear algrebra offers an intuitive explanation. Recall, that for two normalized vectors in $\mathbb R^2$, \begin{equation} \overline v = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}, \enspace \overline w = \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} \end{equation} the inner product is related to the angle between the vectors $\overline v \cdot \overline w = \cos \theta$. Hence the inner product measures the similarity between $\overline v$ and $\overline w$. Alternatively, the inner product $\overline v\cdot \overline w$ expresses the projection of $\overline v$ onto $\overline w$.
This interpretation of the inner product is not restricted to $\mathbb R^2$ but generalizes to other vector spaces. In quantum mechanics a physical state is described by a state vector $| i \rangle$ which lives in some abstract Hilbert space $\mathcal H$. Again, we should interpret the inner product $\langle i|f\rangle$ as measuring the similarity between $|i\rangle$ and $|f\rangle$. Thus, it seems very natural that $|\langle i|f\rangle|^2$ is the probability of finding $|i\rangle$ in the state $|f\rangle$.
While the explanation is by no means a rigorous proof, I find this intuition very useful. I hope it helps you understand the topic.
