Much depends on the kind of sounds (as well as your own skill)
There are several "plug-and-play" options out there. The kind of boards found in toys can be found on eBay and similar for under $10 -- a little more for ones that include amplifier and battery clips in one convenient package. They tend to be limited to a single sound sample and the playback quality is low.
There are slightly more expensive boards; these offer multiple modes (looping, single-shot, etc), and are often available with a built-in amp. Around $20-30 for these. Adafruit has a couple I like the look of a lot; the big advantage is that putting sound files in is drag-and-drop over USB. Someone on this board (sorry, my brain isn't up for names or links this early in my morning) has a couple that have the advantages of being pre-programmed with the necessary sounds and triggers for a number of popular props (aka proton pack, pulse rifle).
The Arduino is extremely powerful for programmability, thus becomes useful if you want a large variety of sounds, complex sounds, or you need the computational power to drive complicated lighting displays, servo-motors, sensors, etc. It isn't the best platform for native sound playback, however. It is just too slow, and the available memory is too small, to really handle playback of sampled audio files. You can do real-time synthesis on the Arduino , however -- my last couple of props with sound used entirely Arduino software synthesis. (The raygun I last did was -- I'm showing off here -- wavetable synthesis on an ATtiny chip!) The programming can be a little hairy though...
The Arduino ecology includes a number of daughterboards that are, basically, like the stand-alone boards mentioned above; drag-and-drop file structure accessed through USB, multiple trigger types, etc. These expand the ability to play back long chunks of high-quality sound, at the addition of footprint and cost (Adafruit's Waveshield is about $40, if I remember). This kind of a package (Arduino and add-on) makes sense mostly in two rather different ways; one is for the power user who really wants all that computational power to filter sensor data and modulate the sounds accordingly (such as, using accelerometers to trigger Iron Man repulsors). The other is for someone who is just getting into this stuff and is willing to bear the cost of a somewhat more pricey system in order to have a relatively painless learning experience. Arduino isn't as plug-and-play as the stand-alone boards mentioned above, but by imbedded microprocessor standards it is training wheels.
I'm going to skip over the ARM ecology, which does offer superior abilities in real-time synthesis and other audio manipulation (at the expense of a steeper learning curve) to close the end of my little graph here with Raspberry Pi. Pi is a full-power computer, a Linux machine chugging along at near gigahertz clocks speeds. It can handle the heavy lifting of audio playback, mp3 decoding, synthesis, fast Fourier transforms, etc. This is one of the few options, for instance, that is truly multi-timbral. On the positive side, it is about the footprint and price of an Arduino. On the downside, it is a full-blown computer, with all the startup time, deep programming, and BSOD woes that entails.
So much for overview!
What you've got is an in-betweener; a stand-alone sound module that can be sent files digitally, but has both a lower-quality sound format and special formatting needs for which you need to download a translator application. It includes a low-power speaker driver (can handle about the speakers found in small toys and greeting cards), and can be run in stand-alone mode (just press a button and the sound will play). Because it is using external memory it can take pretty huge files, leading to long playback (a possible advantage for some props) and if digitally controlled, can be used to "juke box" multiple sound files (one at a time, of course -- multi-timbralism is extremely hard to get at this price range).
I think since you've got it, you should play with it. Start by hooking up speaker and battery and formatting a sound file to stick on the micro-SD. You'll need a USB-to-microSD adaptor, too. Getting it to work in button mode doesn't look difficult. (A little reading suggests that the devices may be factory preset to either button or MP3 mode so this might turn out more difficult than I describe).
Then you can explore controlling it via Arduino. I don't know what your experience is with Arduino. This is a serial connection and may require a little tinkering. If you haven't worked with the Arduino ecosystem before, then first step recommended is to set up Serial Monitor (from the Arduino IDE) and make sure you are spitting correctly formatted serial data! From your description, I can't tell if you have the add-on, a specific Arduino shield, or a combination board that lumps them both together.
