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.