The apparent problem is that most verses (Genesis 7:4, 7:5, 7:10) indicate that Noah (and his family) entered 7 days before the flood began while Genesis 7:13 seem to indicate that Noah entered the ark on the day of the flood.
There are two reasonable explanations for understanding Genesis 7:13 in a way that harmonizes with the other verses.
Explanation 1: “On that very day” refers to the day that God gave the warning to Noah to enter the ark (see Genesis 7:1-4). Genesis 7:13-15 elaborates that Noah did according to what God had commanded (Genesis 7:5).
Explanation 2: “On that very day” refers to the day the flood began, but the verse is not describing that Noah entered the ark on that day. Instead, it points out that by the time the flood arrived, Noah had already done what God commanded—including entering the ark and bringing in the animals.