Laser induced ultrafast demagnetization in ferromagnetic metals was discovered almost 20 years ago, but currently there is still lack of consensus on the microscopic mechanism responsible for the corresponding transfer of angular momentum and energy between electron, lattice and spin subsystems. A distinct, but intrinsically correlated phenomenon occurring on a longer timescale is the magnetization precession after the ultrafast demagnetization process, if a magnetic field is applied to tilt the magnetization vector away from its easy direction, which can be attributed to the change of anisotropy after laser heating. In an in-plane magnetized Pt/Co/Pt thin film with perpendicular interface anisotropy, we found excellent agreement between theoretical prediction with plausible parameters and experimental data measured using time resolved magneto-optical Kerr effect. This agreement confirms that the time evolution of the anisotropy field, which is driven by the interaction between electrons and phonons, determines the magnetization precession completely. A detailed analysis shows that, even though the whole sample is magnetized in-plane, the dynamic interface anisotropy field dictates the initial phase of the magnetization precession, highlighting the significance of the interface anisotropy field in laser induced magnetization precession.