Here "his" refers to Moab which represents the people of Moab. Alternate translation: "their arrogant speech ... like their deeds" (See: [[rc://en/ta/man/jit/figs-metonymy]])
The abstract noun "speech" can be stated as a verb. Alternate translation: "that he has spoken arrogantly" or "that he has spoken arrogant words" (See: [[rc://en/ta/man/jit/figs-abstractnouns]])