What does "to take someone at face value" mean?

The opposite of "taking at face value" is to look for a hidden meaning or ulterior motives. Are they lying? Is this a trick? "Face value" means that there is nothing more than meets the eye and digging deeper isn't likely to reveal anything interesting.

The derivation of "face value" in this sense most likely has to do with the phrase as used in talking about currency: the value of a coin or bill is directly apparent from the numbers printed on it.

So it means taking the meaning of someone's speech or actions directly. (w/o looking for hidden meanings as the other answerers have said)

To take something at face value means to believe what is being said without looking for deeper meanings or hidden agendas.

Well, I couldn't take this question at face value and had to dig deeper (inspired by @sibbaldiopsis' answer).

There were two U.S. Supreme Court cases, Mandeville vs. Welch, 1820, and Lines vs. Sminth, 1851 that spoke of promissory notes needing to have the words "value received" written upon their faces. Could be some connection here.

The first instance of the exact phrase I could find in print was from Mason's Coin and Stamp Collectors' Magazine, 1870, referring specifically to the face value printed on postage stamps (in contrast to the value of the stamp to a collector).

The first instance of the term used for money I found was from Folsom's Logical Bookkeeping, 1873:

Currency too—the bank-note and greenback—has its face-value, which is its printed amount .

The first figurative use of the phrase I found, the most common way it is used today, was from O. F. Cook's 1891, Papers on Myriopoda: