What the extra time and money bought, besides headaches and heartaches for the project’s sponsors and the thousands of patrons who line up for discount theater tickets every day, is nothing less than a new way of seeing the Times Square “bowtie,” that dazzling intersection of Broadway and Seventh Avenue. That is because the TKTS booth proper is topped by a sweeping cascade of 27 ruby-red structural glass steps, rising to a height of 16 feet 1 inch above the 47th Street sidewalk, where hundreds of people (as many as 1,500 if they squeeze in tight) will be welcome to congregate every day until 1 a.m.
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."
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.
Proper is used as adjective here, meaning "belonging or relating exclusively to."
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”.
See the Wikipedia article “Post-positive adjective”.