Why do we say "to be a laughing stock"?

Stocks were a form of public humiliation used as a form of punishment a few hundred years ago.

To answer the first part of your question, laughing is the obvious choice. When you add the -ing suffix to form an adjective, you denote that the sentiment present in the word is transmitted to other people. An interesting book is a book that arouses interest in people. Therefore a laughing-stock makes people laugh at someone.

As for the second part of your question, according to Etymonline, this term was formed by analogy with whipping-stock. See the full entry here.

EDIT upon comment: I'm going to use the examples you give in your question.

An interesting book: a book that arouses the sentiment of interest to the readers.

A cutting remark: a remark that causes the feeling of pain to the receiver.

A boring person: a person that causes the feeling of boredom to other people.

All of the above adjectives could not have the meaning they carry with the -ed ending. Neither could laughed for the same reason. We have, however, interested readers who have the feeling of interest which a book arouses. We also have bored people who have the feeling of boredom that someone else causes (the analogous to cutting can't be used here as it is a different word). Someone who is a laughing-stock makes other people laugh at them. That's what the -ing ending shows.

I'll use the "something solid that things can be fixed to" definition of stock, which Merriam-Webster comes close to agreement with.

1 b archaic : a log or block of wood

In "laughing stock", laughs are the thing which are attached to the stock. "Laughing" in this context isn't the gerund; it's the present participle being used as a noun modifier.

OED says laughing here is a verbal noun, and stock is a verb - which isn't intuitive to me, but...

laughing-stock - an object of laughter; from laughing + stock (v. 2), where...

stock (v. 2) - to strike with the edge or point of a weapon. (cf stock-sword, rapier)

The etymology of this "stock" [F. estoc, It. stocco, prob. of Teut. origin] seems only distantly related to the OED main entry (n.1) [OE. stoc, OFris. stok tree-trunk, stump], where...

stock (n.1) (subentry:8 pl. An obsolete instrument of punishment, which itself derives from subentry:6 a post, stake (the reference being to the two side-posts of the apparatus).

So although it's tempting to see laughing-stock as alluding to the medieval public humiliation device, in fact it's probably more accurate to see it as a metaphorical reference along the lines of bursting someone's bubble of hypocrisy/pomposity with the sharp-pointed weapon of laughter (perhaps akin to poking fun at).