The evidence for fluidity comes from experiments. One such experiment is hybridisation of cells; this is where two cells are fused together.
An hour after fusion, the red and green dyed proteins have dispersed evenly across the membrane. This shows that the proteins move laterally within the membrane.
Different experiments involve dyeing either phospholipid or protein molecules, or attaching fluorescent markers and bleaching an area. Over time the bleached areas ‘disappear’, as the bleached and marked molecules mix.