This is the oldest blessing still in regular use, given to Aaron to speak over the people. It has three lines, and each one moves closer.
The Lord bless thee, and keep thee. Blessing and keeping together — provision and protection, the giving and the guarding of what is given. It is possible to be given much and kept from nothing; the prayer asks for both.
The Lord make his face shine upon thee, and be gracious unto thee. A shining face is the ancient way of describing pleasure at seeing someone. It is what happens involuntarily when a person you love walks into a room. The blessing asks that God look at you like that — and then, immediately, that he be gracious, because we all know we have not earned the look.
The Lord lift up his countenance upon thee, and give thee peace. To lift up the face toward someone is to give them full attention. Not a glance in passing; a turning toward.
God's comment on the blessing is the last line: they shall put my name upon the children of Israel; and I will bless them. The priests speak it, but the blessing is not theirs to generate. They are naming what God intends to do.