If cutting is involved, line up the old tile and use a good pair of scissors to cut the vinyl.
If no cutting is involved, then place the new vinyl tile in the grid. In our basement, we chose to keep the old tile up there for soundproofing reasons.
Add the new ceiling tile back into the grid.
Optional: You can cover your current grid with this product.
Video
Notes
There are other brands of drop ceiling tiles you could try.
They come in different designs and sizes to fit your drop ceiling grids.
Our ceiling has not sagged at all with leaving the old tiles.