Stochastic variation in protein expression generates phenotypic heterogeneity in a cell population, and has important role in antibiotic persistence, mutation penetrance, cancer growth and therapy resistance. Studies investigating molecular origins of noise have predominantly focused on the transcription process. However, the noise generated during the transcription process is further modulated by the translation process, which eventually determines the expression noise at the protein level. Studies across different organisms have revealed a positive correlation between translational efficiency and protein noise. However, the molecular basis of this correlation has remained unknown. In this work, through stochastic modeling of translation in single mRNA molecules and empirical measurement of protein noise, we demonstrate that ribosome demand associated with high translational efficiency drives the correlation between translational efficiency and protein noise. We also show that this correlation is present only in genes with bursty transcription, where ribosome demand varies with stochastic fluctuations in mRNA numbers. Thus, our work reveals how transcriptional bursts are translated into protein noise, which has important implications for investigating protein noise and phenotypic heterogeneity across biological systems.