Q:

How are 'marry', 'merry', and 'Mary' pronounced differently?

The way I pronounce these words is the same. Similarly for other words like these: I pronounce ferry and fairy the same, carrot and caret. Yet, dictionaries show different pronunciations for these words:
For example, Merriam-Webster gives:
\ˈma-rē\ for marry, \ˈmer-ē\ for merry and \ˈmā-rē\ for Mary
The American Heritage Dictionary gives:
(măr′ē) for marry, (mĕr′ē) for merry and (mâr′ē) for Mary
When I listen to the recordings, they all sound the same to me. How do you pronounce them differently? Who pronounces them this way?

A:

Interestingly, this question appeared as number 15 on the Harvard Dialect Survey, so it is possible to give a good summary of the pronunciation differences in these three words as they are spoken in the United States.
The 11,422 respondents were asked to choose from five options given the following prompt:

How do you pronounce Mary/merry/marry?

The percentages next to each choice indicate the proportion of survey participants that chose that option:

a. all 3 are the same (56.88%)
b. all 3 are different (17.34%)
c. Mary and merry are the same; marry is different (8.97%)
d. merry and marry are the same; Mary is different (0.96%)
e. Mary and marry are the same; merry is different (15.84%) 

The maps that show where in the US the respondents for each answer choice were located are available here, but can be summarized by noting that mid-westerners (residents of Ohio, Illinois, etc.) seem most likely to pronounce all three the same, and New Englanders seem most likely to pronounce at least two of them differently.
As for non-US speakers, a linguistics blog post by Ryan Denzer-King claims that,

…in rhotic dialects, intervocalic resonants tend to be ambisyllabic, i.e., they are attached both to the syllable that precedes them (as a coda) and the syllable that follows them (as an onset). An /r/ in coda position tends to neutralize many if not all vowel quality distinctions in the syllable it closes, and thus in rhotic dialects, where these syllables are closed by an /r/, we get all three front vowels neutralized to the [-hi][-lo][+ATR] vowel /e/. For non-rhotic speakers, /r/ can never be in coda position, and thus this neutralization does not occur.

So it is more likely that you would hear people talk without these three sounds being merged into one in places outside of North America. This seems logical to me, but I don't know enough rhotic speakers to say if it matches my experience, and there will of course be exceptions everywhere.

A:

If you wish to try to simulate the distinction for people who do not normally make it, I have found that it is best to illustrate it this way:

Marry has the same vowel as Matt or mat, so IPA /æ/.
Merry has the same vowel as met, so IPA /ɛ/.
Mary has the same vowel as mate or may, so IPA /eɪ/ or /e/, depending on just how glide-y you are feeling.

Not that almost anyone still does that, but some do. Once you have that down, you can practice saying “I’ll marry merry Mary” so that all three sound different using the schema given above.
As far as actual length goes, the vowel of the middle one is the shortest, or most clipped. J.R.R. Tolkien pronounced the name of his character Merry in this clipped fashion, but he also had the old-style r often enough. 
For a living example, you can hear Larry King pronounces his own name Larry, which would rhyme with to marry, not with Mary.  He indeed uses the vowel of lag, mat, hat there. It sounds really weird if it is not in your own accent, as I can myself attest.

A:

Cameron's excellent answer shows that most people in the US do indeed pronounce these words the same. Although I see that you are a native US English speaker, I'd thought I'd contribute the British English version.
In British English, these words typically sound distinctly different. Anecdotally, I think the difference is fairly consistent across the different regions of England at least.
Since you have already looked up phonetics for these words, I'll try and explain the difference by comparing words with similar vowel sounds that you may also pronounce differently.
Marry: Harry, carry. (Short vowel as in fat, cat.)
Merry: berry, Terry (Short vowel as in get, wet.)
Mary: hairy, fairy (Long vowel as in air, bear.)

