To determine why an ice skater would sink if he/she stood in the sand, let's think about someone walking on snowy ground:
If there is a person who is walking on the snow on the ground, most of the time (unless the snow is really packed) their feet will sink into the snow. However, if someone is wearing "snow shoes," then that person is able to walk on top of the snow. Why is this??
First of all, snow shoes look somewhat like tennis rackets that people strap their shoes to. Then when someone walks on the snow with snow shoes on, their weight is distributed evenly across the entire surface area of the shoes. However, if someone is trying to walk on top of the snow without their snow shoes on, then their weight is concentrated more. This is because the surface area of their feet is a lot less than that of their snow shoes. So all of their weight is pushing down on just their feet instead of being spread out across a bigger area. This ultimately causes this person's feet to sink in the snow.
Similarly, when someone is standing in the sand with their feet, then their weight is distributed evenly across their feet; and they are able to stand on top of the sand. But as soon as someone puts on ice skates, their weight is concentrated to only the surface area of what is touching the sand: the blades of their skates. This then causes them to sink just like a person’s feet would sink in the snow.