For a compact quantum group \mathbb{G} of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L^\infty (\mathbb{G}) into an ultrapower of the hyperfinite II _1-factor (the Connes embedding property for L^\infty (\mathbb{G})). We establish a connection between the Connes embedding property for L^\infty (\mathbb{G}) and the structure of certain quantum subgroups of \mathbb{G} and use this to prove that the II _1-factors L^\infty (O_N^+) and L^\infty (U_N^+) associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N \ge 4. As an application, we deduce that the free entropy dimension of the standard generators of L^\infty (O_N^+) equals 1 for all N \ge 4. We also mention an application of our work to the problem of classifying the quantum subgroups of O_N^+.