The mother probably won't relax until she hears that her son arrived safely.

Literal

Mother [topic-は] son [subject-が] safely arrived [quotative-と]-[nominalizer-いうこと] [object-を] hear [until-まで] [topic-は] be-relieved not-will [probably-だろう]

Deeply nested: ということを聞くまでは安心しないだろう. The embedded report (息子が無事ついたということ) is the object of 聞く, まで+は limits the scope: 'until she hears... she won't.' だろう adds speculation. 無事 = safely/without incident.