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We investigate the dynamics of a magnetoelastic robot with a dipolar magnetic head and a slender elastic body as it performs undulatory strokes and burrows through water-saturated granular beds. The robot is actuated by an oscillating magnetic field and moves forward when the stroke amplitude increases above a critical threshold. By visualizing the medium, we show that the undulating body fluidizes the bed, resulting in the appearance of a dynamic burrow, which rapidly closes in behind the moving robot as the medium loses energy. We investigate the applicability of Lighthill's elongated body theory of fish locomotion, and estimate the contribution of thrust generated by the undulating body and the drag incorporating the granular volume fraction-dependent effective viscosity of the medium. The projected speeds are found to be consistent with the measured speeds over a range of frequencies and amplitudes above the onset of forward motion. However, systematic deviations are found to grow with increasing driving, pointing to a need for further sophisticated modeling of the medium-structure interactions.