Meaning
Bajrang comes from the Sanskrit vajra anga, limbs of thunderbolt, and bali from balin, the strong one, so the cry hails Hanuman as the hero whose body is unbreakable. The epithet recalls the story of Indra's vajra striking the child Hanuman as he leapt at the sun: the blow broke his jaw, which is how he came to be called Hanuman, and only afterwards, when the gods appeased his father Vayu, did Indra's own boon make his body harder than the thunderbolt. Jai means victory or hail, so the whole phrase is less a request than a shout of allegiance to that strength.
बजरंग बली की जय।