Ah "The Scottish Play"

Many of the special effects were done just as they are today, with harnesses, ropes, trapdoors, "magic" and light. One of those effects was how the Globe burned down originally (gunpowder from a cannon wadding caught the theater on fire). Shakespeare also used words to describe the scene as narration so the audience would imagine what was happening in the background instead of seeing it as you would in a film.
The Globe was split into three sections; The main stage, the Heavens above the stage and Hell below the stage. The heavens were used to fly in gods/goddesses and other heavenly creatures and trapdoors in the floor were used to raise demons and dead people from under the floor.

The heavens had the fly system with harnesses to lower actors onto the stage by the use of weights and pulleys. Its the same way Peter Pan flies today, just less safe. The cannon was store there for booming effects and thunder was made by either banging drums or waving a sheet of metal depending on the depth of sound needed. You could also do lightning with the use of gunpowder or use chemicals to produce different colors. There is rumor that the witches comment in the play about "the fog and filthy air" is a nod to the use of sulfur which smells horrid, being used to create the fog while they are casting their spell for *whispers* Macbeth.

The main stage is where the most of the action happened. Stabbings and blood were done with bladders filled with animal blood and poked at the right time to leak on the costume to show that someone was wounded. The narration that set the scene here happened as well, and the magic. Like magicians today, sleight of hand and tricks were used to show beheaded heads speaking (the body of the actor would be in a trapdoor in the Hell section of the stage and his head on the top, or a contraption would be used where one actor would have his head inside a box on the opposite side of a second actor who had his head exposed to speak). Fake bodies on turntables to show gruesome disembowelment using animal parts were also possible.

Firecrackers were a favorite when devils came up from Hell, to spark and bang (also were used for battlefields), and different actors would use their voices to mimic animals needed like roosters or dogs.

A lot of special effects in theater still have the same basic principals now. There is still a harness flying system and trapdoors, and the use of practical special effects and audience distraction to bring them about.

Because one can never have too much of a good thing, I’ll also recommend this answer from u/cdesmoulins:

• The Globe theatre burned down when a cannon was fired during a performance of Henry VIII, did Elizabethan actors often use effects of this kind?