It refers to the blasting of plants by the sun during high summer. Etymologically, it’s the result of being struck by a star, as it comes from the Greek astron (as in astronomy and many other words), plus bolis, a missile (which is also the source of bolide).
The star is Sirius, the dog star; because it rises and sets with the sun during summer in the northern hemisphere, it has lent its name to dog days for the hottest part of the year in places north of the equator. The dog days are those from about the middle of July to the middle of August (though the exact dates vary depending on where you live).
The thought behind astrobolism is connected to the old idea that this period of summer is under a malign influence, in which dogs run mad, the air is unwholesome, sunstroke is common, and all useful works stagnate for want of effort.
It was first recorded in Nathaniel Bailey’s Dictionary, dated 1721. Apart from very occasional appearances in other reference works, it has had almost no circulation at all.