Why did Japanese invade the Solomon islands?

Answer 1

To shore up their flank against the U.S. Navy.

Basically, the causation follows the general expansion of defensive perimeter following Japan's invasion of islands further North and West.

This was in order to protect it's land campaigns in Asia from Allied forces across the pacific.

All of this, in turn, was in order to eventually control the sources of coal in China and Southeast Asia, and the eventual conquest of that region.

Wikipedia covers it in more detail with material excerpted from noted printed texts.