What does it mean to have a dimension of $1.5$?

How do I explain to my dad what it means for some object to have a dimension of 1.5?

Show him the Minkowski sausage fractal curve. It has a fractal dimension of exactly 1.5.

Fractional (or fractal) dimensions are used for shapes that have a perimeter or area that is fractal in some way. In fractals, the shape gets more and more complex as you zoom in to it. A good example is a coastline: depending on how closely you follow the bays, inlets, puddles, grains of sand and even molecules, the measured length of the coastline changes. The smaller the detail, the longer it gets.

For example, to construct the Koch curve, you start with a line segment. The central third of it is replaced by 2 segments at 30 degrees. Then each of the shorter segments is manipulated in the same way, and so on at infinitum. No matter how far you zoom ito the Koch curve, it will always look the same. As a result, the distance between any 2 points on the curve is infinite, and the Koch curve has a dimension or 1.262.

Start by having a look at Wikipedia's "Fractal dimension" page.

As @GSmith said in a comment, the Minkowski Sausage has a dimension of 1.5. To see what it looks like, try Wolfram Alpha and enter the number of iterations you would like (it starts with 3). Using 0 iterations, the shape is a square. As the number of iterations increases, the length of the curve grows without bounds while the enclosed area remains the same as the square (1).