bigger animals just have more cells. You see, an important part of cells is maintaining a specific surface area to volume ratio. This ratio changes when the cell gets bigger or smaller, and that disrupts several life processes, so similar cells are usually similarly sized.
Although cells vary in size, they are mostly close to the same (microscopic) size. larger animals just have a LOT more of them. There are a few outliers which are single celled animals which are visible to the naked eye!