。
She kept talking to her friend even after class started.
Literal
She [topic-は] class [subject-が] started [even though-ても] friend [to-に] continued-to-talk.
~ても (even though) sets up the contrast — class started, but she didn't stop talking. 話し続けた uses the compound verb suffix ~続ける (to continue doing), attached to the masu-stem of 話す. 友達に話す uses に to mark the person spoken to. The sentence paints a vivid classroom scene of someone obliviously chatting away.