The lever rule gets its name from a physical analogy to a simple lever (seesaw). Just like a heavier weight on the shorter side of a lever can balance a lighter weight on the longer side, the phase at the shorter end of the tie line will have a higher phase fraction than the phase at the longer end. This is just an analogy that helps you remember which side will have the higher phase fraction. Even though the math ends up being the same for physical levers and for the phase diagram lever rule, the physical meaning of the variables is completely different.

It also helps to think about what happens as you move from a single-phase region into a two-phase region. Just before crossing the boundary, only one phase is present. Just after crossing it, only a small amount has transformed into the new phase — so the original phase is still dominant. As you move further into the two-phase region, more and more transforms, until you approach the other boundary and the new phase dominates. This makes it intuitive why the phase on the shorter side is more prevalent: you haven't moved far from where it was the only phase present.