A lamellipodium is the result of rapid polymerization of actin filaments pushing against the membrane while they are stabilized into branches by proteins like Arp2/3. The protrusions then adhere to the surface using integrins. The integrins, while attached to the surface, are also connected into actin inside the cell.
Then, the myosin and actin at the cell body-lamellipod junction contract, pulling the cell forward while cell adhesion sites at the tail of the cell come apart. Sometimes, the adhesions at the tail may be too strong and will result in a portion of the cell membrane getting left behind.
Cells move by creating protrusions of their cell membrane called lamellipodium.