While geniice covered the rockets, the "bombs bursting in air" refers to mortars fired from the mortar ships HMS Terror, Volcano, Meteor, Devastation, and Aetna. Rather than the mortars in use today, which are predominately lightweight constructions used as light artillery by infantry units, the mortars of the day looked more like this and fired what were essentially hollow cannonballs filled with black powder and with black powder timefuses which gunners would preset before firing, and the unreliable fuses of the era meant that they could explode high in the air, long after hitting the ground, or even not at all. Generally, however, the intent would have been to explode shortly above the ground in an airburst for greater casualty producing effect.

/u/ArguingPizza more or less covered it. There was a great variety of explosive ordinance in the early 19th century, and the British had access to just about all of it on the Atlantic seaboard. One detail not covered in Pizza's post was that the mortars fired high, arcing shots that were designed to go over the walls of fortifications and explode among "soft" targets behind the walls. Flat-trajectory fire was meant to batter walls and dismount artillery, although the advantage in 1814 was nearly entirely on the side of the fortifications, which were designed to slope flat fire off of a glacis and to enclose and protect its defensive artillery in fixed, hardened bastions.

At Fort McHenry, the British idea was more "shock and awe" than destruction. The high, lofting fire of the mortars and the screech and scream of the rockets was intended to drain the morale and the will to resist from the garrison, or, failing that, to convince the defenders in Baltimore that the fort would be swallowed and overcome in fire. A constant barrage and trading fire would engulf the area in smoke, and soon only flashes of light would be able to be seen.

The morale effect of that smoke clearing and the US flag still flying high was, as you can imagine, dramatic and uplifting to US forces. Hence the song.

Strategically, the fort standing prevented the British from using their fleet to its best effect during the attack on Baltimore, and allowed the Americans to concentrate their defenses without having to split forces to thwart British landings elsewhere around the city.

But yes, watching an artillery bombardment through the war, the smoke and vapor trails of the rockets, the sizzling and screaming of the falling mortars, the flash, crack and rolling booms of the cannons and their answering explosions when the bombs and case shot exploded would have been extremely dramatic. Key was certainly amplifying the effect in the Star Spangled Banner, but not by much.