بالأحرى = more correctly speaking / more exactly / more clearly
لاحظ خط اليد, أو بالأحرى خطوط اليد = notice the handwriting, or more exactly, the handwritings.
اليوم سنقدم درسا, أو بالأحرى شرحا لقواعد اللغة العربية = Today we'll present a lesson, or more clearly, an explanation of the Arabic grammar rules.
It became more clear, or more exactly, easier = أصبحت أوضح, أو بالأحرى, أسهل.
Hopefully it became clear now, and if you have any questions concerning this explanation, please don't hesitate to ask.