Ummm this question made me think about a lecture from my teacher. When he was teaching us about the Big Bang theory!
From Wikipedia: The big bang states: The Big Bang theory is the prevailing cosmological model that describes the early development of the Universe.1 According to the theory, the Big Bang occurred approximately 13.798 ± 0.037 billion years ago, which is thus considered the age of the universe. At this time, the Universe was in an extremely hot and dense state and began expanding rapidly. After the initial expansion, the Universe cooled sufficiently to allow energy to be converted into various subatomic particles, including protons, neutrons, and electrons. As stated here.
Example of balloon:
So what you can imagine, there is a baloon with some dots on it. When you are blowing it, the balloon gets bigger, its size expands. But the dots stay where they were. There do comes a change in their size too. Galaxies expand too, but that doesn't mean that there is amount being added, but that means there is some space some extra space being added to it.
Objects around you aren't expanding. Because this change would occur in years, so its almost ~0 this size as the changes occur inside the atom, atom contains magnetic field by the movement of electrons that would lesser down expansion and bla bla which is Chemistry so skip this. However, galaxies, stars, and on whole universe is expanding! And its been years it was expanding, and will expand! :)
Your plane example:
And your plane? Imagine that the plane is expanding. Hey wait! try this one (if possible, as this one would explain it fairly), place some magnets over a magnetic field, make sure north faces the north so they keep floating. Now when the surface expands, the magnets would stay there! They won't move backward or forwards just as the universe (plane) is moving. Similarly, as universe expands, planets don't expand! As they are not stitched to the universe or clamped to the universe. They are free moving.