독송하다 = 1) to recite out loud, 2) to recite in order to memorize the text.
It is not a very common word.
The most common word for reciting text is 낭송하다 (e. 시 낭송 = poem recitation).
음송하다 is another less common word.
송 means 'to recite' in all these. The first character details the meaning: 독 = to read, 음 = to recite/sing out loud, 낭 = to recite in a clear, sonerant voice.