In Fullmetal Alchemist, it is stated a number of times that a state alchemist's pocket watch increases their power, but there doesn't seem to be a clear mechanism for that. How does that work, and what does increasing the power of an alchemist mean?
As stated in this wiki,
...each State Alchemist is given a silver pocketwatch to hold as identification. This watch is engraved with the symbol of the State Alchemist program - the Amestrian Dragon circumscribed in a hexagram (though both anime series alter the hexagram to a nondescript polygram, presumably so as not to invoke Judaism). Although the manga does not describe the pocketwatch as anything more than official proof that the carrier is a State Alchemist, the 2003 anime suggests that each pocketwatch is an alchemical amplifier. The 2003 anime also depicts the watch's silver chain as extendable, as in the instance when the Strong Arm Alchemist, Major Alex Louis Armstrong uses it to tie up a target.
So, according to the 2003 anime, the watch amplifies the alchemist's powers. However, both manga and FMA:Brotherhood shows that the watch is simply a form of identification, much like a police badge.
Also note that in the anime the watch is the indicator which people use to recognize Edward as a State Alchemist. I believe that unless the State Alchemist himself tells you who he is, the presence of the watch is the only other way to know it (they also get a certificate, but I doubt they carry it around).
They never explained it because it is a development of the first anime adaptation that was never expounded upon. The watches were merely forms of identification as state alchemists in Brotherhood, which follows the manga more closely.
They actually showed this in one episode when Ed got his watched repaired, that it contained red rocks in the back of it and that was how they amplified their alchemic powers.
The Silver Pocketwatches amplify the State Alchemist's power, but besides that, the only thing they really do is prove that you're a State Alchemist, and perhaps to tell the time.
They tell time but mainly symbolize being a state alchemist and part of the country's government and army if necessary.
In the 2003 anime the silver watch has some kind of incomplete philosopher's stone that amplifies the transmutations
Some of the information contained in this post requires additional references. Please edit to add citations to reliable sources that support the assertions made here. Unsourced material may be disputed or deleted.