Drano is made of lye (sodium hydroxide), mixed with little bits of aluminum metal.
When lye crystals mix with water, they react strongly, and produce a lot of heat. The hot water softens lumps of grease in the drain. The sodium hydroxide reacts with the softened grease, turning it into soap. The bits of aluminum metal also react with the sodium hydroxide, releasing more heat, but also releasing a lot of hydrogen gas, which makes bubbles that further loosen the fat and grease in the clog.
The hot water dissolves the soap. The bubbles of hydrogen break up the lumps of grease and make them float up away from the clog, so more hot water and lye can react with the grease and fat.
If there is hair in the clog, the lye also dissolves that, and the hydrogen bubbles loosen the strands from one another, so that the clog catches more water and is carried down the drain.
So there are many chemical reactions going on at once. Heating the water, making soap, making bubbles, and dissolving proteins. Add to this the force of new water pushing on what remains of the clog, and lye-based drain cleaners do a pretty good job.
Lye can cause burns to skin and eyes, so caution should be used with any product that contains sodium hydroxide.