What is a theming bonus?

Many of the wonders and buildings you can place Great Works into have slots for more than one work. In such buildings, there is an optional Theming Bonus that will give you extra base tourism; usually +2 in a building with two slots, +3 in a building with three, and +4 in a building with four.

Each building/wonder with a theming bonus available has different requirements to earn it. Sometimes, you need to have all the works be from the same civilization and era, sometimes all from different civilizations and eras, sometimes it may be something as complex as "Must have two Great Works of Art and two Artifacts, all of which are from different eras and civilizations".

You can see all your theming bonuses from the new Culture Overview screen, in the "Your Culture" page which lists all your Great Works and their locations. To see the requirements you'll need to meet for any building that has more than one slot, hover over the large number to the right of the slots; that number is your current theming bonus, and hovering over it should bring up a tooltip with the details needed to get the bonus.

A quick browse through the Civilopedia lists the following as having more than one slot:

• Museums have two Art slots. Get +1 for having either both items be your civ, or both from another civ. The other +1 is for having both from the same era. Works with either Great Works of Art, or Artifacts, but not one of each.
• Hermitage has three Art slots. +3 for having all three slots be Great Works of Art from different eras and civilizations.
• Broadway has three Music slots. +3 for having all three be from the same era and civilization.
• Oxford University has two Writing slots. +2 for having both be from the same era, and from civilizations that are not Oxford's owner (can match each other or be different civilizations).
• Sistine Chapel has two Art slots. +2 for having both be Great Works of Art from the same era and civilization.
• Louvre has four Art slots. +2 for having two Great Works of Art and Two Artifacts, all four from different eras and civilizations.
• Uffizi has three Art slots. +3 for having all three be Great Works of Art from the same era and civilization.
• Globe Theatre has two Writing slots. +2 for having both be from the same era and civilizations.
• Great Library has two Writing slots. +2 for having both be from a different era and civilization.
• Sydney Opera House has two Music slots. +2 for having music from two eras, but the same civilization.

Here is a chart showing the various theming bonuses for the slotted Great Works:

Source: /r/civ

There's nothing wrong with the current answers, but they are lacking a bit in why theming bonuses are so important. After all, it looks like an awful lot of work to get some of them. Part of the question is how theming bonuses work. Part of that is why they work.

Theming bonus multipliers.

All over the game, there are opportunities to increase tourism. At least two of those opportunities are directly about the theming bonus.

• French (Napoleon): Museum and World Wonder theme bonuses are doubled in their capital.
• Aesthetics: Adopting all policies doubles the theming bonus you receive from Museums and Wonders.

Which effectively means the theming bonus for the Sistine Chapel and the Louvre can be increased to 8 and 16 respectively.

Theming bonus is just one of the bonus methods for increasing tourism, but a potentially very important one. Put on it on top of:

• Buildings, like Airport (50% from culture of wonders and improvements in the city added to its tourism output, +50% tourism output of Great Works in the city)
• Wonders, like National Visitor Center (100% of the culture from wonders and improvements is added to the tourism output of the city, +100% tourism output for Great Works in the city)
• Technologies, like Internet (doubles tourism output)

and those theming bonuses can help you get a hefty amount of tourism. Theming is not a multiplier, it's a bonus before the multipliers start doing their work. All those tourism multipliers multiply your theming bonus just like they multiply the tourism from 'ordinary' Great Works.