Biopolymer translocation is a key step in viral infection processes. The transfer of information-encoding genomes allows viruses to reprogram the cell fate of their hosts. Constituting 96% of all known bacterial viruses [A. Fokine and M. G. Rossmann, Molecular architecture of tailed double-stranded DNA phages, Bacteriophage 4, e28281 (2014)], the tailed bacteriophages deliver their DNA into host cells via an "ejection" process, leaving their protein shells outside of the bacteria; a similar scenario occurs for mammalian viruses like herpes, where the DNA genome is ejected into the nucleus of host cells, while the viral capsid remains bound outside to a nuclear-pore complex. In light of previous experimental measurements of in vivo bacteriophage λ ejection, we analyze here the physical processes that give rise to the observed dynamics. We propose that, after an initial phase driven by self-repulsion of DNA in the capsid, the ejection is driven by anomalous diffusion of phage DNA in the crowded bacterial cytoplasm. We expect that this two-step mechanism is general for phages that operate by pressure-driven ejection, and we discuss predictions of our theory to be tested in future experiments.