Preliminaries Recall that a representation of an algebra on a Hilbert space is a map from the algebra to the bounded operators on a certain Hilbert space. Also recall the Heisenberg canonical commutation relations $$[q_i,p_k]=i\delta_{ik}I$$ A representation of such relations is a set of operators on some Hilbert space that satisfy the same commutation relations. A typical example is the Schrödinger representation, which in one dimension is realised by the multiplication operator $q=M_s$ and the differentiation operator $p=-i\frac{\text d}{\text ds}$ on the Hilbert space $L^2(\mathbb R)$ with Lebesgue measure.
The equivalence of the different pictures is a consequence of von Neumann's uniqueness theorem, which states that every irreducible representation of the Heisenberg's uncertainty relations is unitarily equivalent to the Schrödinger representation. So if you start with Heisenberg's matrix mechanics, i.e. you assume that you have a representation by (infinite) matrices of the canonical commutation relations, then there exists a unitary that "translates" the action of these matrices over some Hilbert space into the action of the Schrödinger operators $q$ and $p=-i\hbar\nabla_q$ on the Hilbert space $L^2(\mathbb R^n)$ with Lebesgue measure. The place where these two different pictures meet is the Dirac notation involving bras and kets. One way of actually proving this is a la Dirac-Dixmier, which involves studying the spectrum of the quantum harmonic oscillator and proving that the Hamiltonian is essentially self-adjoint as a consequence of Nelson's criterion.
The rough idea behind the result is that uniqueness of the Schrödinger's representation follows roughly from the fact that the Weyl algebra stemming from the Weyl's operator is isomorphic to the C*-algebra of compact operators on a infinite-dimensional separable Hilbert space, which is known to only have one class of unitary equivalence of irreducible representations. This can be constructed using the magic positive type function on the Heisenberg group $$\phi(z,t)=e^{-\frac{\Vert z\Vert^2}4+it},\qquad(z,t)\in\mathbb C^n\ltimes\mathbb R.$$