I was relieved to hear that you arrived safely.

Literal

Safely [adverbial-に] honorable-arrived [quotative-と] heard [and] felt-relief.

おつきになる is the sonkeigo (honorific) form of 着く ('to arrive') — お + つき + になる. 安心する ('to feel relieved, to feel safe') takes no particle for the object of relief. Warm expression of concern followed by relief.