Why are bras and kets defined to be Hermitian adjoints of each other?

For a single particle the wave function is given by some map $psi in L^2$, the dirac notation is just a simple notation which uses the vector space properties of $L^2$ since the scalar product is defined to be $$ (psi_1 , psi_2 ) = int _{mathbb R} psi_1(x)^* psi_2(x) text{d} x $$ Therefore if we write down the scalar product of two states described by their wave function to be $$ langle psi_1 | psi_2 rangle. $$ If you look at the definition of the scalar product it therefore makes total sence to "define" $$ langle psi | ^dagger = | psi rangle $$

Let us understand this in the context of finite vector space, i.e one where the vector space dimension 'n' is finite.

You can understand the problem with an analogy to a Complete Orthogonal Real vector space (with identity as metric), in short the one which we usually study in elementary high school. You can ignore the complications of completeness, etc for time being. Consider all the vectors originating from the origin. Now these vectors can be imagined as column matrices. A general inner product is defined as $bar{v}^T bar{w}$ and a particular type of this inner product, called norm is given as $bar{v}^T bar{v}$. A norm is a mapping from vector space (containing all v) and its dual vector space (containing all $v^T$) to real number line i.e. you input a vector and its dual and get a real number as an output i.e. $bar{v}^T bar{v}$= real number. Here 'T' denotes a transpose of the column matrix v. The elements of these vectors as defined earlier are also just real numbers. Hence, it just suffices to take a transpose, as product of two real numbers is just a real number. Physically, the square root of this real number output after taking the norm denotes the distance of the endpoint of this vector from the origin, i.e. $sqrt{(a-0)^2+(b-0)^2+(c-0)^2}$, where the endpoint p = (a,b,c) and origin=(0,0,0).

But, in Quantum Mechanics, which is bit more generalized than the above case, i.e. the elements of the column matrices now can be complex numbers. We need to generalize this norm now to ensure that the mapping to real number line is preserved, because if we use the same definition of norm as earlier then we can easily see that we will have product of complex numbers. A product of complex numbers is never a real number until and unless it is multiplied with its' complex conjugate, and nothing else. Hence, we include the condition of complex conjugation in the above definition of norm to ensure that the output received is a real number. Hence, the norm is now defined as $bar{v}^daggerbar{v}$, where $dagger = *T$ where *T denotes a combined operation of complex conjugation and transpose.