。
As soon as the call connected, she started talking.
Literal
She [topic-は] phone [subject-が] connected [as soon as-とすぐ] talk began.
~とすぐ ('as soon as') uses the conditional と to mark a trigger event, combined with すぐ ('immediately'). 電話が通じる means 'the phone connects / gets through.' 話し始めた uses ~始める ('begin to'), one of the aspectual compound verbs. The sentence captures the urgency of someone who couldn't wait to speak.