High-resolution radar range profile (RRP) is crucial for accurate target recognition and scene perception. To get a high-resolution RRP, many methods have been developed, such as multiple signal classification (MUSIC), orthogonal matching pursuit (OMP), and a few deep learning-based approaches. Although they break through the Rayleigh resolution limit determined by radar signal bandwidth, these methods either get limited super-resolution capability or work well just in high signal to noise ratio (SNR) scenarios. To overcome these limitations, in this paper, an interpretable unfolded neural network for super-resolution RRP (DSSR-Net) is proposed by integrating the advantages of both model-guided and data-driven models. Specifically, DSSR-Net is designed based on a sparse representation model with dimension scaling, and then trained on a training dataset. Through dimension scaling, DSSR-Net lifts the radar signal into high-dimensional space to extract subtle features of closely spaced objects and suppress the noise of the high-dimensional features. It improves the super-resolving power of closely spaced objects and lowers the SNR requirement of radar signals compared to existing methods. The superiority of the proposed algorithm for super-resolution RRP reconstruction is verified via experiments with both synthetic and measured data.