Attempt at answering 'why does this work' question.
This formula assumes the cells linked to these three checkboxes are on the same row for each student. A checkbox writes '1' into a cell if it's checked, and '0' if it's not checked. Therefore, the total for all 3 checkboxes is at least three if all three are checked.
The "row()" part of the formula makes it more flexible since we don't know which rows you're using. The formula row() returns the current row number and row() & ":" & row() creates a string that selects the entire row (such as 4:4 means all cells in row 4).
The countif() part generates a result only if a given condition is met (that's the IF part).
What I like to do with these nested formulas is break each part down and write it out on its own line so you can see what is going on.
You could also do a simple SUM() and check if it = 3, but that depends on how the checkbox output cells are used on the sheet.