。
As soon as she saw me, she started running toward me.
Literal
She [topic-は] me [object-を] sees [when-と] immediately my direction [toward-へ] began-running.
私の方へ literally 'in my direction,' uses 方 (ほう, 'direction, side') with the directional particle へ. へ emphasizes the direction of motion (vs に, which highlights the endpoint), making it a natural choice when the focus is the trajectory rather than the arrival. 走り出す pairs 走る ('to run') with the sudden-onset auxiliary ~出す ('burst into') — the moment-of-starting compound. Combined with ~とすぐに ('as soon as'), the whole sentence captures a sudden, impulsive run toward someone the moment they're spotted.