The idiom or saying of Empty as last years bird nest means that a home that parents are living in is empty of the children that they raised and loved in their home.
The saying comes from when birds have baby birds and hatch those eggs and then later those birds leave the nest.
Just like when you have kids and then later on in years the kids grow up into adults and then leave the home and leaves an empty nest.
Parents who loved their children dearly can feel depressed when they have an empty home after all those years of having those wonderful children.
They hate to see their children leave their nest which is the home they reside in and have raised those children up in.
The empty nest refers to A family home inhabited by parents after their children have grown up and departed.