The term "apodizing filter" is actually somewhat vague, and refers to several related types of filters, so it should always be used carefully.
In the more general sense used in digital audio, an apodizing filter is simply a filter that alters the sound in a certain way.
More specifically, the majority of apodizing filters are intended to reduce or eliminate pre-ringing in the digital audio signal.
Filtering is required at several points in the conversion process from analog to digital and from digital to analog.
The filters typically used produce a certain amount of ringing - both before and after the original signal.
However, different types of filters produce different amounts of this ringing, and may distribute more or less of it before or after the original sound.
So, if you send a short sharp sound through the system, when it comes out, there will be some ringing added before and after the sound; this is an unavoidable part of the process.
While it may seem that, intuitively, a filter with no ringing at all would be best, that is simply not an option (the only way to have a filter with no ringing is to serious compromise its performance in other ways - which is audible).
Whether this ringing is audible and, more importantly, whether some sorts of ringing are more audible than others, or more "unnatural sounding", has been disputed for quite some time.
One major camp holds that, because it never occurs in nature, pre-ringing is especially audible and annoying, while post-ringing is rather innocuous.
What most "apodizing filters" do is to, by using some really fancy math, "shift pre-ringing into post ringing".
They alter the signal in such a way that any pre-ringing that is present is reduced or eliminated - but at the cost of adding more post ringing.
The typical net result of an apodizing filter is that the pre-ringing is gone, but the amount of post-ringing is usually increased by a slightly larger amount.
(Since the pre-ringing was "annoying", but the post-ringing is "innocuous", this works out to be a net improvement in the overall sound quality.)
NOTE:
1) that there are lots of ways of doing this - and it can be done to varying degrees
2) apodizing filters often ALSO alter other things, like frequency response, which can make it difficult to figure out what you're hearing
(Do those guitar strings sound cleaner because the pre-ringing has been removed, or simply because the high end is also rolled off by 3 dB by the particular filter you're using?)
Part of the MQA "process" is that, as part of the encoding process, the MQA encoder "alters the signal to remove ringing or time smear that was introduced when it was originally converted".
Another part of the process is that it specifies that the DAC use certain specific filters when playing back that content (presumably they've chosen filters that preserve the improvement rather than eliminating it).
The confusing part is that these two things have been tied together in the MQA specifications and process.
(You COULD apply the same corrections they do, but not use their recommended filter; and you COULD use a filter that's the same as the one they recommend with any file you like.)
So, if you play that file through "all MQA stuff", you will have the MQA correction, played back in the MQA approved fashion, using an MQA approved filter.
And, since it's all part of the MQA process, and your MQA products will know how to handle it, you don't have to figure out what to turn on and off along the way.
The MQA decoder doesn't "tell your processor" anything... it simply plays back MQA encoded content using the settings it was programmed to use.
(There is a presumption that, with MQA encoded content, the encoder has already "figured out what corrections to apply" and applied them.)
However, to answer your question in a general way, many DACs offer a selection of multiple filters... (that includes our Big Ego and Little Ego DACs)
And, of those that do, one of the options most offer is quite similar to the one used by MQA... (usually called "a reduced pre-ringing filter" or "an apodizing filer")
I've never seen a DAC that would "decide" which filter to use; you're expected to pick the one that sounds best to you.
I should also note that this sort of processing CAN be done purely in software... although, depending on the filter used in your DAC, it might get "undone" by the DAC itself.
As a broad generality, many separate high-end audiophile DACs offer the choice between several filters...
So, yes, they offer you a choice of whether to use an apodizing filter or not (possibly even a choice between multiple different ones).
But pre/pros generally do not; most pre/pros simply use what their makers consider to be the most accurate filter available...
Most processors already have an extensive and detailed list of features...
At that level, choosing what filter to use in the DAC is on a level of detail with "choosing a custom color for the stitching on the floor mats in your new car".
(It's considered to be a really fiddly detail that most people wouldn't bother with.)
According to some research that's been published recently, the filter characteristics chosen for use with MQA actually perform poorly in certain other ways...
(Which means that you may not want to use them with non-MQA content anyway.)
So, it seems like one of the main advantages of MQA is it can tell your DAC/processor if the audio samples already reflect apodization of audible frequencies. Your processor can then decide whether or not to apply apodization before it goes analog. Do any processors out there have a apodizing switch you can turn on or off when you already know if your source material has been through that? Sounds like a pain, but could it be significantly cheaper than paying for the MQA pipeline?