Short answer
The Qur'an says clouds are driven gently, joined together and piled into a heap. Rain comes from their midst, and hail falls from mountain-like masses in the sky, with lightning bright enough to dazzle. It also calls the clouds heavy. Every part of that is true. A single ordinary cloud holds about 500 tonnes of water, and it floats because its drops are tiny. The Prophet Muhammad (ﷺ) lived in a desert town and could not read. The learned world of his day explained storms with Aristotle's theory of dry vapours catching fire. The Qur'an repeats none of that theory. It describes what anyone can see in a storm and offers no theory of how it works. The Qur'an calls such things signs of God (﷾).
Full explanation
The storm in one verse
The Qur'an gives the life of a thunderstorm in one verse. Clouds are driven, joined and heaped up. Rain comes from their midst. Hail is sent down from mountain-like masses. Lightning is bright enough to blind. The Qur'an presents all of it as the work of Allah (﷾), a sign to reflect on.
“Seest thou not that Allah makes the clouds move gently, then joins them together, then makes them into a heap? - then wilt thou see rain issue forth from their midst. And He sends down from the sky mountain masses (of clouds) wherein is hail: He strikes therewith whom He pleases and He turns it away from whom He pleases, the vivid flash of His lightning well-nigh blinds the sight.”
What meteorologists find
Thunderstorms grow as cumulus clouds (heaped clouds) rise and merge into a towering cumulonimbus, the tall storm cloud. It is often ten kilometres high or more. Hail forms in its cold upper part, carried up by strong updrafts (rising air). The charge behind lightning builds as ice and soft hail collide up there. The Arabic speaks of mountains in the sky with hail in them. The translation renders this as mountain masses of cloud. The towering storm cloud with hail in its upper part fits this closely.
Explains that thunderstorms grow from rising, merging cumulus clouds; that strong updrafts carry raindrops into the very cold upper part of the storm, where hail grows; and that the electric charge behind lightning builds up as small ice particles and soft hail (graupel) collide in that cold upper region.
“Seest thou not that Allah makes the clouds move gently, then joins them together, then makes them into a heap? - then wilt thou see rain issue forth from their midst. And He sends down from the sky mountain masses (of clouds) wherein is hail: He strikes therewith whom He pleases and He turns it away from whom He pleases, the vivid flash of His lightning well-nigh blinds the sight.”
The heavy clouds
Twice the Qur'an calls rain clouds heavy (thiqal, meaning laden). Measurement bears the word out. An ordinary fair-weather cloud holds about 500 tonnes of water, and a thunderstorm cloud holds far more. It floats because its water is spread as tiny droplets through a huge volume of air. So the cloud is lighter than the drier air below it. No one in the seventh century could weigh a cloud, and the Qur'an's word fits what was later measured.
“It is He Who doth show you the lightning, by way both of fear and of hope: It is He Who doth raise up the clouds, heavy with (fertilising) rain!”
“It is He Who sendeth the winds like heralds of glad tidings, going before His mercy: when they have carried the heavy-laden clouds, We drive them to a land that is dead, make rain to descend thereon, and produce every kind of harvest therewith: thus shall We raise up the dead: perchance ye may remember.”
Works out the weight of the water in an ordinary cumulus cloud from its size and the amount of water per cubic metre, and arrives at roughly 500 tonnes (about 1.1 million pounds). It explains that the cloud still floats because its water is spread as tiny droplets through about a cubic kilometre of air, which makes the cloud less dense than the drier air beneath it.
What the learned world taught
The science of the age was Aristotle's. He was a Greek thinker of the fourth century BCE. He taught that the air holds two vapours from the earth, one moist and one dry. Thunder was the dry vapour striking the clouds, and lightning was that vapour catching fire. His Meteorology (his book on the weather) was not put into Arabic until the Abbasid translations (the translation movement under the caliphs of Baghdad). That was well over a century after the Prophet (ﷺ) died. His weather theory was still taught in Europe in the 1600s. Greek learning did circulate in the region in Syriac, in the monasteries and schools of Syria and Mesopotamia. Syriac is a form of Aramaic used by the eastern churches. So anyone who drew on it would have picked this theory up with it.
Aristotle teaches that the sun draws two exhalations from the earth: a moist one, vapour, and a dry, smoke-like one (Book I, ch. 4). In Book II, ch. 9 he explains thunder as the dry exhalation, trapped in a cloud and forced out, striking the surrounding clouds, and lightning as that same ejected exhalation burning with a fine, faint fire.
The standard history of the Graeco-Arabic translation movement. It shows that the translation of Greek science and philosophy into Arabic was an Abbasid project centred on Baghdad, running from the second half of the eighth century to the tenth, with Greek works earlier available in the region mainly in Syriac.
Why this is a sign
Watching storms is open to anyone, and the Qur'an does not claim otherwise. The sign is in what is said and what is left out. The Prophet (ﷺ) could not read or write, and the Qur'an records it. Its account of the storm says a few plain things that are true. It uses a word for the clouds that measurement bears out. It carries none of the errors every learned source of the age would have handed on. The Qur'an promises that such signs will keep appearing until its truth is clear.
“And thou wast not (able) to recite a Book before this (Book came), nor art thou (able) to transcribe it with thy right hand: In that case, indeed, would the talkers of vanities have doubted.”
“Soon will We show them our Signs in the (furthest) regions (of the earth), and in their own souls, until it becomes manifest to them that this is the Truth. Is it not enough that thy Lord doth witness all things?”
Perspectives
The Islamic perspective
The Qur'an, revealed to the Prophet Muhammad (ﷺ), describes the storm accurately, calls clouds heavy, and avoids the false theory of its age. It offers these as signs of the One who drives the clouds, not as a science lesson.
The Qur'an describes clouds driven, joined and heaped up. Rain comes from their midst, and hail is sent down from mountain-like masses in the sky, with blinding lightning.
Thunderstorms form as cumulus clouds rise and merge into tall towers, and hail grows in their cold upper part.
The Qur'an twice calls rain clouds heavy, and an ordinary cloud does hold about 500 tonnes of water.
The standard learning of the age, Aristotle's, explained thunder and lightning as a dry vapour striking the clouds and catching fire. The Qur'an repeats none of it.
The Qur'an states that the Prophet (ﷺ) did not read or write before it came.
The Qur'an gave an accurate account of the storm, free of the errors of its age, to a man who could not read. This is one of the signs the Qur'an says God (﷾) shows in the world.