This is the very definition of the inner product using bra-ket notation. You could change that equals sign into a "is defined as" sign, because this equation is defining the meaning of the left-hand side of the equation.
In a finite-dimensional complex Hilbert space, the bras and kets represent vectors (kets being traditional column vectors of the complex components of the vector, one component for each dimension; and bras being the same, but a complex conjugate transpose [i.e. a row vector of the complex conjugates of the dimensional components]).
Hence, for a finite number of dimensions, n, you can think of bra-ket notation as simple matrix algebra: a 1-by-n bra times an n-by-1 ket results in a scalar 1x1. In regular old 3-dimensional space, this is the same as the dot product. For example, given vectors A and B,
<A|B> = A^*_xB_x+A^*_yB_y+A^*_zB_z= A_xB_x+A_yB_y+A_zB_z =A \cdot B
where Ax is the x-component of vector A, etc. Note that, since we're talking about real 3-dimensional space, the complex conjugate is meaningless, and hence the relationship boils down to the dot product.
Now, in the context of quantum mechanics, we're no longer talking about a finite-dimensional space. The "dimensions" in this case may represent possible quantum states, of which there can be infinitely many (though most states usually have near-zero probability).
The relationship stays the same; it's equivalent to taking the dot product of two vectors in n-dimensional (complex Hilbert) space as the number of dimensions goes to infinity. The only difference is that, due to the infinite dimensionality, the summation changes to an integral.
I hope that helps.