What do swellings on the dorsal roots of the spinal nerves signify?

Nerve cell bodies. The dorsal root ganglion (the enlargement) is the place in the nerve where the cell bodies of feeling nerves (sensory nerves) are located.
Irritation. If the nerve roots are under pressure from a disc herniation or tightness from the spinal canal itself ( i.E stenosis), the nerve roots will be compressed and they become inflamed and swelling occurs. This is painful and the treatment can be exercises, injections and ultimatley surgery to relieve pressure if non surgical methods to not work.