It is an interesting question, and I am not completely sure on everything I write below.
First of all, atomic volume is defined as the "quotient obtained by dividing the atomic weight of an element by its specific gravity". (Merriam-Webster) It is interesting to note that I could find only a few publications that have the term "atomic volume", and most of the references are from 1980's. So, it seems that the use of atomic volume has almost disappeared from mainstream chemistry, except perhaps a few niche areas.
The definition essentially means that you measure the volume of one mole of the substance, and then divide it by Avogadro's number to get the atomic volume. So, that means measuring atomic volume just requires a way to measure the volume of a fixed weight of substance. Perhaps Lothar Meyer did something like that. Atomic volumes can also be calculated from the packing density, if you have a crystalline element.
But anyway, let's look at the trend you showed in the question. The atomic volume is the largest in the alkali metals and smallest in the middle of the period. From Chemistry of the Elements1:
Fig 2.1 shows a modern version of Lothar Meyer's atomic volume curve: the alkali metals appear at peaks and elements near the centre of each period (B,C; Al, Si; Mn, Fe, Co; Ru, and Os) appear in the troughs. This finds a ready interpretation on the electronic theory since the alkali metals have only 1 electron per atom to contribute to the binding of the 8 nearest neighbour atoms, whereas elements near the centre of the period have maximum number of electrons available for bonding. ... Closer inspection reveals that a much more detailed interpretation would be required to encompass all the features of the curve which includes data on solids held by very diverse types of bonding. ...
So basically the simplistic explanantion boils down to the number of unpaired valence electrons each element has. Remember that the atomic volume is the volume of one mole of element divided by a constant. The volume of one mole of element means that we are considering the volume of a fixed number of atoms i.e. it depends only on how close the atoms can pack in the solid state, and does not depend on mass.
As the number of unpaired valence electrons is the highest in the middle of a period, the packing is strongest there and the atomic volume is lowest. It's likely that the trend is atomic radius becomes negligible compared to this effect in solid state.
- Chemistry of the Elements, N. N. Greenwood, A. Earnshaw, Butterworth-Heinemann, 1998, p 23-24