끓이다 is bringing something to the boil or making it by boiling, and the clause in front is the step just before: mixing the sugar and water and boiling it, measuring the water out and starting it boiling, drawing hot water and boiling that, putting rice in water and boiling it into porridge. Two lines are about who does the boiling rather than how, asking someone to come over and cook it for them. 끓다 is the water boiling by itself.