How are spherulitic structures formed in basalt rock?
According to the image that you provided, what you are seeing are not spherulites, but rather epoxy bubbles. More on that later. I will first answer what are spherulites:

Spherical features observed in thin sections can form by several ways.


Devitrification - this is the recrystallisation of glass. I think it is more common in rhyolitic rocks than in basaltic rocks. As the glass is a non-stable amorphous phase, there is a solid state transition of the glass to quartz or feldspars. The centre of the spherulite is usually some kind of an disturbance in the glass. Could be a crystal nuclei or a small bubble. Here is a magnificent example of a spherulite in rhyolite, from the beautiful collection of Bernardo Cesare:





Filling of amygdules - this usually happens in more basic rocks, for example basalts. A rounded cavity is left due to the presence of gas bubbles during solidification. In this case, the minerals grow from the walls of the cavity inwards, filling the empty space (in contrast to the inside to outside growth pattern of spherulites). The minerals that grow are usually zeolites, clays, quartz and various other hydrothermal minerals. I remember seeing a magnificent example of epidote filling an amygdule once in a basalt. Here's an example of an amygdule from the Imperial College Rock Library:



Notice how it doesn't completely fill in the gas cavity.

These are the two most common ways in which this can form. However, in your case I think the answer is less exciting than this. I am pretty sure that your rounded objects are simply bubbles in the epoxy. Here's why:


They are perfectly rounded. This rarely happens naturally.
They are completely empty.
The actual rock is to the left of that image. Most of the area covered in your image does not seem to have any real rock in it, unless it is glassy but I doubt it because there is rock on the left.
That 4-sided halo around it occurs almost entirely in artificial bubbles created during the sample preparation process. You can put it a λ plate and see that opposite sides have similar colours (either red or blue).