Translation:At that place, readers always laugh.
Yes, it means a place in the book.
Yes, you can definitely be at a particular place in a book - I just sat here thinking about it going "I've lost my place" and other various versions of it, and it definitely works.
I think probably I personally might be more inclined to use page, but I think that's pretty much personal preference, I can't think of an expression where 'place' doesn't work just fine.
And what about "part"? would the sentence "the readers always laugh at this part" sound natural?
It first bears noting that that the English translation here doesn't really seem natural. The idea of "at that place" should go after "laugh," in keeping with the common distinction in English and Russian word order in expression actions in a place.
However, "Readers always laugh at that place" sounds much more like a reference to the physical location (e.g. coffee house, park, living room) where they are reading, sort of an odd statement (are we talking about a clubhouse for laughing readers?). It really ought to be "Readers always laugh at that place in the book." On the other hand, to me at least (AmE speaker), "Readers always laugh at that spot" carries the intended meaning of "point in the plot" and sounds completely natural, just a bit less formal than "place in the book."
Ну, да. А где еще? Там обычно бывают и читатели и книги смешные. Вот и смеются там.
It could be but needn't be. The dividing lines between этот/тот and this/that are pretty different. Тот is used a lot less than "that."