بدأ = started
عهد = era
الحمدين = name of this era '' its just a name''
and its a dual name ( for example hamad is the original name, hamadeen that means two of hamad)
في مهمتهما = in their task ( because its a dual name)
التي كلفا بها = which they were assigned to it
so the meaning here
and the HAMADEN era started in their Mission which was assigned to them