Why would a tender offer be less than the market price?

Rational shareholders wouldn't accept such an offer. The company making the offer is simply trying to get a deal with questionable — though not illegal — tactics.

Your answer is actually in the link you provided. Quoting from the fourth paragraph [emphasis is mine]:

The SEC has cautioned investors about mini-tender offers, noting that "[s]ome bidders make mini-tender offers at below-market prices, hoping that they will catch investors off guard if the investors do not compare the offer price to the current market price." The SEC's tips for investors regarding mini-tender offers may be found on the SEC's Web site at http://www.sec.gov/investor/pubs/minitend.htm.

There's a lot more information at the SEC web page mentioned. One part I'll highlight from the SEC web page mentions another reason for a mini-tender:

Investors need to scrutinize mini-tender offers carefully. Some bidders make mini-tender offers at below-market prices, hoping that they will catch investors off guard if the investors do not compare the offer price to the current market price. Others make mini-tender offers at a premium – betting that the market price will rise before the offer closes and then extending the offer until it does or improperly canceling if it doesn't.

You'll also find a lot more information at Wikipedia - Mini-tender offer.

As an addition to Chris Rea's excellent answer, these tender offers are sometimes made specifically to cast doubt on the current market price. For instance, a large public company that contracts with a smaller supplier or service company, also public, might make a tender offer below market price. The market will look at this price and the business relationship, and wonder what the larger company knows about the smaller one that they don't.

Now, what happens when investors lose confidence in a stock? They sell it, supply goes up, demand goes down, and the price drops. The company making the tender offer can then get its shares either way; directly via the offer, or on the open market.

This is, however, usually not successful beyond the very short term, and typically only works because the company making a tender offer is the 800-pound gorilla, which can dictate its own terms with practically anyone else it meets. Such offers are also very closely watched by the SEC; if there's any hint that the larger company is acting in a predatory manner, or that its management is using the power and information of the company to profit themselves, the strategy will backfire as the larger company finds itself the target of SEC and DoJ legal proceedings.