Basically, within a 4cm by 4cm square, you must ensure the centre of the disc lies within the centre 2cm by 2cm square that is a subset of the 4cm by 4cm square.
Recognise that the diameter physically restricts the area in which the centre of the disc can fall. The squares against the boundary are effectively trimmed by 1 cm because the radius is 1 cm, and the centre of the disc cannot utilise the 1 cm in which the radius of the disc occupies space.
It's a 5x5 board with dimensions 20cm x 20cm. However, due to the 1 cm trim along the boundaries, the total area of the board in which the centre may land: 18cm x 18cm = 324 cm squared. That's your sample space.
It's easier to think of the disc as a "point" and only consider its physical size (a 2 cm diameter disc) for working out things like: "which area can the point lie in, so that the whole disc is in the square" and for the boundaries, where we trim off 1 cm because of the disc occupying space.