The phrase implies heavy repetition, similar to how a drumbeat is consistent and persistent. The coach drummed the strategies into the players until they could execute them perfectly.
Pro Tip 2/3
Authoritative Tone
'Drum into' often suggests an authority figure is doing the teaching. The sergeant drummed the importance of discipline into the new recruits.
Pro Tip 3/3
Mild Frustration
It might imply mild frustration at the need for such repetition. I have to drum into him the importance of being punctual for our meetings.