People carefully remembering things so that they can think about them later is spoken of as if they were putting those things safely in their hearts. Alternate translation: "kept them in mind" or "thought carefully about these matters" or "thought a lot about these events" (See: [[rc://en/ta/man/translate/figs-metaphor]])
"What kind of great person will this baby grow up to be?" It is also possible that this question was meant to be a statement of their surprise at what they had heard about the baby. Alternate translation: "What a great man this child will be!" (See: [[rc://en/ta/man/translate/figs-rquestion]])
The phrase "the hand of the Lord" refers to the Lord's power. Alternate translation: "the Lord's power was with him" or "the Lord was working in him powerfully" (See: [[rc://en/ta/man/translate/figs-metonymy]])