Interplay of spin, charge, orbital and lattice degrees of freedom in oxide heterostructures results in a plethora of fascinating properties, which can be exploited in new generations of electronic devices with enhanced functionalities. The paradigm example is the interface between the two band insulators LaAlO3 and SrTiO3 (LAO/STO) that hosts two-dimensional electron system (2DES). Apart from the mobile charge carriers, this system exhibits a range of intriguing properties such as field effect, superconductivity and ferromagnetism, whose fundamental origins are still debated. Here, we use soft-X-ray angle-resolved photoelectron spectroscopy to penetrate through the LAO overlayer and access charge carriers at the buried interface. The experimental spectral function directly identifies the interface charge carriers as large polarons, emerging from coupling of charge and lattice degrees of freedom, and involving two phonons of different energy and thermal activity. This phenomenon fundamentally limits the carrier mobility and explains its puzzling drop at high temperatures.