What does "proper" mean in this context?

Solution 1:

It means the booth specifically, without any extra bits. By way of example: "Times Square" might often be used to refer to the area around Times Square, but may include things which are not actually part of the Square. To narrow such a usage, one might say "I mean only the actual Times Square" or "I mean Times Square proper."

Solution 2:

It's definition 6 from Merriam-Webster:

6 : strictly limited to a specified thing, place, or idea

So it means on the TKTS booth itself, and not on the pavement or anywhere else.

Solution 3:

Proper is used as adjective here, meaning "belonging or relating exclusively to."

Solution 4:

You can read “the TKTS booth proper” as “the TKTS booth, strictly speaking” or “the TKTS booth itself”.

We sometimes position an adjective after the noun it modifies. In that sentence, proper is an adjective modifying booth, and means “strictly speaking”. Other examples: “proof positive” and “knight errant”.

References

See the Wikipedia article “Post-positive adjective”.