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This is done by STOSD instruction that stores a dword value from EAX into a memory location which address is in EDI register ("D" means destination). After the value from EAX is transferred to memory the instruction increments EDI by 4 and EDI register now points to the next DWORD in memory if DF flag is 0. If DF flag is 1 then EDI value is decremented by 4 and EDI now points to the previous DWORD in memory.
If we prefix any instruction with REP prefix it causes the instruction to be repeated until the value in ECX register is decremented to 0. For example we can write very simple code that should theoretically zero "all memory" (practically it will trap because of access violation):