How can you repair and why would ground erode under cement stairs?

I'd put small stone under the step, to fill it in. You can also mix the stone with cement, and then fill it in. The stone should be the size of a walnut or larger.

Could re-fill with soil or Concrete or lay paving slabs around step.

A possible explanation is the soil has eroded because the roots of the grass have been killed and there is nothing to hold the dirt in place, causing it to be washed away and the structure has shifted significantly .

Another thought is the water table in your area has changed over time and with it raising and lowering, the earth underneath your steps responded by moving.

You could try raising it professionally and having a solid layer put under it, such as stone or concrete.

Water or drainage issue. Do you have gutter and where do they drain or run off. Water always wins

This is common for newer homes where the ground settles, but that doesn't look to be the case here. The stairs look to be in place for some time already. Filling the void is probably the best route. Not filling it will result in critters finding their way under the steps and making it their home. Additionally, you now have water draining toward your structure instead of away. In the past, I've used sand, soil or a combination of the two to push under the steps to raise the height of the ground.

If by 'drain tile' you're talking about what we call weeping tile, there would be no concern as it's meant to be under the ground anyway, so putting more soil on top shouldn't be an issue.

I agree with Redcactec in that the stairs may need to have a solid base instead of just being on the ground in order to prevent further erosion.

Looks like a low sot and water has been just eroding the soil. I would fill the area with concrete/stone mix. Mix the concrete to a thicker consistency. Use a hoe or fabricate a pushing stick and stuff the concrete into the crevice. Smooth it out on the outside. Once it hardens use some soil around the low spots and slope it away from the steps packing it down firmly. The grass will eventually fill in.

Animals digging underneath can cause erosion as well.