From my very brief skim of the field, it seems like the consensus is that savants have access to the kind of low-level information processing which non-savants do not. I'll summarize one such theory in some detail, since it's the one that I've happened to read. But I'm not an expert in this field and this is just one of the theories, the rest of which I haven't studied. I'll provide some pointers to the other theories at the end.
Snyder (2009) speculates that "savants have privileged access to lower level, less-processed information, before it is packaged into holistic concepts and meaningful labels". This is something that everyone would in principle have the potential for: savants are accessing information that is present in everyone's brain, but not normally available for introspection. As some supporting evidence for this idea, Snyder notes that savant-like abilities can sometimes be artificially induced in healthy individuals by inhibiting a part of the brain, and that savant abilities are known to appear spontaneously and suddenly in childhood or as a result of brain injuries. Once acquired, savant abilities typically do not improve by training.
I have argued that the extraordinary skills of savants are latent in us all and that they can be induced artificially owing to the inhibiting influence of low-frequency rTMS, that is, by turning off part of the brain, not by exciting it. My hypothesis is that savant skills are facilitated by privileged access to raw, less-processed sensory information, information that exists in all brains but is inaccessible owing to top-down inhibition. Thus, autistic savants tend to see a more literal, less filtered view of the world. Their ‘skill’ or performance does not depend on active learning, but simply on an effortless ‘reading off’ of this less-processed information.
Snyder does not explicitly draw the connection between this and memory, but I would infer the connection to be as follows. Non-savants mostly consciously process information that has been packaged into holistic, high-level concepts, and it is these high-level concepts that are stored into memory. When we try to recall something that happened, we cannot recall specific details, because they were never stored into memory in the first place. Savants think more in terms of low-level details, so that's also what gets stored in memory.
As for the autism connection, Snyder proposes that a lack of high-level concept formation and seeing things mainly in terms of low-level detail is what autism essentially is. High-level concepts allow us to order the world and understand it, and autists are lacking in those kinds of concepts.
So why are all autists not savants? There are many varieties and degrees of autism, and it is conceivable that a lack of high-level concept formation could be caused by many different things. Some of those things could also cause more widespread damage to other systems. Snyder's suggestion is that autist savants represent autism in its "purest" form, uncontaminated by other disorders. (And why are all savants not autists? Presumably because they have access to both low-level and high-level concepts.)
How well-supported is this theory? Writing in the same issue as Snyder, Happé & Frith (2009) note that Snyder's hypothesis is "controversial", but note that a lot of other theorists also propose explanations which seem to imply some sort of focus on low-level processing:
Several authors in this issue offer hypotheses for why autism should be associated with talent, with remarkable consensus that the ability to process local information plays a key role. Happé & Vital (2009) suggest that detail-focused attention and memory predispose to the development of talent, both in the general population and in autism. Baron-Cohen et al. (2009) suggest that superior sensory acuity across modalities underlies such detail focus, which in turn fosters the tendency to explore and master closed systems (e.g. the calendar). With the enhanced perceptual functioning theory, Mottron et al. (2009) propose that locally orientated processing and, specifically, detection of patterns in the environment, underlies the high incidence of savant skills in autism. Plaisted Grant & Davis (2009) also emphasize the qualitative differences in perceptual and cognitive processing in ASD, and make tentative links to underlying neural systems. These very different theorists share the view that an ability to attend to and process featural information plays an important part in predisposing to special skills of a savant sort. This view is upheld, for example, in a detailed case study of a prodigious mnemonist and calculator with Asperger syndrome and synaesthesia who shows a preference for local processing and an unusual pattern of brain activation while remembering digits (Bor et al. 2007).
Likewise, Neumann et al. (2010):
One way to approach the question of how savants achieve their
astonishing abilities is to investigate their brain activity during
savant skills demonstration. Most theoretical accounts act on the
assumption that some lower-level (perceptual) information processing
mechanisms are enhanced in contrast to [41,7], or at the
cost of higher-level operations [65,20,56,16]. Thus, single aspects
of perception or lower-level information processing are preferentially
processed or accessed resulting in a local or early processing
bias that is benefiting for the development of savant skills. This is
associated with an enhanced memory trace for these features .
Treffert , assuming different brain mechanisms, postulated a
model that implies compensatory right-hemispheric functioning
after left-hemisphere damage and reliance upon lower-level procedural
memory due to damage of higher-level memory.
So at least some version of this theory seems to be right. See the references below if you want to look more into the other theories.
Happé, F. & Frith, U. (2009) The beautiful otherness of the autistic mind. Philos Trans R Soc Lond B Biol Sci 364(1522):1345-1350. http://rstb.royalsocietypublishing.org/content/364/1522/1345.full
Neumann N, et al. The mind of the mnemonists: An MEG and neuropsychological study of autistic memory savants. Behav Brain Res (2010). http://ki.se/content/1/c6/10/06/19/Neumann_2010_BBR%5B1%5D.pdf
Snyder A. (2009) Explaining and inducing savant skills: privileged access to lower
level, less-processed information. Philos Trans R Soc Lond B Biol Sci 364(1522):1399–405. http://rstb.royalsocietypublishing.org/content/364/1522/1399.long
References cited in the quote from Happé & Frith:
Baron-Cohen S., Ashwin E., Ashwin C., Tavassoli T., Chakrabarti B. (2009) Talent in autism: hyper-systemizing, hyper-attention to detail and sensory hypersensitivity. Phil. Trans. R. Soc. B. 364, 1377–1383. doi:10.1098/rstb.2008.0337.
Bor D., Billington J., Baron-Cohen S. (2007) Savant memory for digits in a case of synaesthesia and Asperger syndrome is related to hyperactivity in the lateral prefrontal cortex. Neurocase. 13, 311–319. doi:10.1080/13554790701844945.
Happé F., Vital P. (2009) What aspects of autism predispose to talent?. Phil. Trans. R. Soc. B. 364, 1369–1375. doi:10.1098/rstb.2008.0332.
Mottron L., Dawson M., Soulières I. (2009) Enhanced perception in savant syndrome: patterns, structure and creativity. Phil. Trans. R. Soc. B. 364, 1385–1391. doi:10.1098/rstb.2008.0333.
Plaisted Grant K., Davis G. (2009) Perception and apperception in autism: rejecting the inverse assumption. Phil. Trans. R. Soc. B. 364, 1393–1398. doi:10.1098/rstb.2009.0001.
References cited in the quote from Neumann et al.:
 Bölte S, Hubl D, Dierks T, Holtmann M, Poustka F. An fMRI-study of locally oriented
perception in autism: altered early visual processing of the block design
test. J Neural Transm 2008;115(3):545–52.
 Happé F. Understanding assets and deficits in autism: why success is more
interesting than failure. Psychologist 1999;12:540–6.
 Snyder AW, Mitchell DJ. Is integer arithmetic fundamental to mental processing?:
the mind’s secret arithmetic. Proc Biol Sci 1999;266(1419):587–92.
 Mottron L, Dawson M, Soulieres I, Hubert B, Burack J. Enhanced perceptual
functioning in autism: an update, and eight principles of autistic perception. J
Autism Dev Disord 2006;36(1):27–43.
 Treffert DA. Extraordinary people. Lincoln, NE: iUniverse, Inc.; 2006.
 Waterhouse L. Speculations on the neuroanatomical substrate of special talents.
In: Obler LK, Fein D, editors. The exceptional brain.NewYork: The Guilford
Press; 1988. p. 493–512.