Piezo processors come in two primary flavors. The L.R. Baggs Para Acoustic DI is a good example of the first one, straight up tone shaping. The unit is basically a combination of a preamplifier stage like you'd find on a combo amp, running through a DI transformer to produce the balanced lo-Z output. The Fishman Aura Spectrum is an example of the second and much more extreme solution, digital modelling, specifically "acoustic impulse response" modelling.
The preamp solution is 100% about using EQ to smooth out the harsh plastic tone. That's a combination of the preamp's and transformer's natural "coloration" of the piezo tone, and your ability to dial in more or less of that color by tweaking the EQ.
The exact native EQ curves of these are proprietary, but you can explore what the units do with an external soundcard and some frequency-analysis software (available free in a variety of flavors, personally I'd try Room EQ Wizard, built to assist in "ringing out" performance and recording spaces). Set up the DI in a "loopback" configuration, where the output of the interface feeds the input of the DI and the output of the DI feeds back to the interface. Now, feed the device pink noise, and compare the output spectrum of the noise (should be a straight line at the output level on a logarithmic frequency chart) with the input spectrum of the unit to see how the unit colors the input signal with the controls in any configuration you like.
Now, while these units can help, a lot of the problem is that a piezo really isn't the wonder pickup its touted as. You'll hear a lot about how the piezo was designed to capture the total instrument, responding to vibrations throughout the instrument body, not just the strings. That's not really true. What it was actually designed to do is disappear into the traditional styling and construction of the acoustic guitar, changing its native acoustic characteristics as little as possible. One critical problem with piezos is that they capture the vibrations of the string at an anchor point. Every guitar player knows that the closer to the bridge that you strum or pluck, the harsher the tone gets, and that pickups get brighter but more nasal the closer to the bridge saddles you place them, as the fundamental tone is reduced in favor of higher overtones. Well, the piezo is sensing the vibrations in the worst possible place in terms of brassy, nasal tone. Preamp-based solutions can only do so much; they can help, but they can't add the parts of the waveform that you're not getting because of how the piezo senses the instrument, and so you just can't EQ that plastic tone out completely.
Enter the second major solution these boxes bring to the party; digital modeling. The Fishman Aura Spectrum and others like it use a technology originally developed for digital reverbs; an "impulse", a short burst of white or pink noise, is played into a resonant space, while a reference microphone captures the space's reverberation of that impulse sound. This "impulse response", more or less an acoustic fingerprint of the space's reverb time across the frequency spectrum, can be analyzed and applied to any other sound, making that sound appear to have been played in that resonant space.
It worked so well for modeling acoustic environments that other people tried applying it to other things we commonly try to mimic, like guitar amplifiers and speaker cabinets, and eventually, a methodology for capturing the impulse response of acoustic instruments themselves was developed. You can now load these acoustic guitar IRs into an IR loader like the Fishman Aura Spectrum, and the DSP unit will take the direct signal of the piezo and apply the resonant qualities of the modeled guitar from the IR. The result is not 100% perfect, but it's 1,000,000% better-sounding than your typical piezo under-saddle pickup running totally direct.
The biggest problem is it's not your guitar's tone you're hearing anymore. Between the tone changes of the piezo and the IR, what you hear through the speakers really can't be described as the sound of the guitar itself. It's a good sound, but it's not the sound you paid to get when you bought the guitar. You might as well be playing the cheapest s**tbox you can find hanging off the pawnshop wall. Some people care about that, others don't. Personally, I like what my guitar sounds like, it's why I bought it, and I want my amplification system to give me more of it. The silver lining is that there are IR profiles of most of your common guitar models, so you can load an IR that may not be your exact guitar, but is one of the same or a very similar model. There are also other units that have the IR capture process built-in; you set up a microphone of your choice in front of the guitar, plug that mic and the guitar into the effects box, and the effect will capture the difference in signal between the direct input and the actual sound of the guitar through the mic, creating a profile very like an IR that serves the same purpose.
There are other options. One of the best-liked is an internal microphone; instead of a piezo sensing the vibrations of the string relative to the bridge through the saddle, you literally put a contact microphone into the soundbox of your guitar, and it senses the internal resonance of your instrument through the air. Virtually no plastic tone (from a good system), and it's your actual guitar you're hearing with little or no DSP, very similar to playing in front of a mic on a boom stand pointed at the neck-body joint. The primary downside is gain before feedback; because the microphone directly senses the air in the soundbox, if the soundbox is pointed at any part of the PA system that's open to that guitar's sound, the speakers can induce vibration in the soundbox, and it'll feed back just like an open mic. There are preamp systems designed to split the difference, incorporating both a piezo and an internal mic and allowing you to dial in an acceptable blend while rejecting feedback, but these are among the most expensive systems you can retrofit into a guitar.
Soundhole pickups, basically a specially-voiced version of the magnetic inductor coils you can find on any Strat or LP, are another option.
Other solutions center on the bridge material. The mentality is that if the guitar sounds like plastic, at least part of that is because you're probably capturing the strings' vibration through plastic. Replacing your nylon, acrylic or Tusq saddle with an actual bone saddle can help, but it will also have a more marked effect on your guitar's unamped acoustic tone that you may not like (especially if your guitar is already naturally bright).
At the end of the day, anything other than sitting in front of an unamped acoustic guitar and listening to it is going to be a compromise. The nature of the compromise you are willing to live with, not only in sound but in the appearance of the instrument and of course your budget, are going to determine which one works best. However, in pretty much any live sound scenario, you're not going to be able to get the absolute most honest, accurate reproduction of your guitar at any price, because the realities of the venue just won't allow it.