The most likely function of these spheres is recreational. The psionic circuitry stimulates various centres of the brain. The effect is similar to hallucinogenic drugs. This is the only evidence that aliens have any cultural or recreational pastimes.
These craft components, although maybe interesting to look at, serve no function other than to increase your funds. There is no reason why these should not be sold off immediately. You can make a reasonable sum of money with them, as many Alien Bases, the Terror Ship and the Battleship contains a numerous quantity.
What actually constitutes Alien Entertainment has baffled many as the UFOPaedia entry specifically states that they are "spheres". The only object which resembles this (see pic to the right) is the "bloodshot eyeball on a stem" (there are also other names for this) but its MCD record shows it is just junk. Also, note that the eyeballs *only* appear on ships which have alien entertainments, and*always* appear in these cases. The only things which are Alien Entertainment are the 4 wall segments shown above.
This could merely be a "low-res" problem. If you look closely, you can see that the orange walls closely resemble the orange background. If you look even closer, you can see various colored dots on the walls which might be the spheres.
For each wall tile you recover, you get the same amount back at base (for a 1:1 ratio). Each nets you a cool $20,000, so be careful.