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I'm writing a book about humans finding alien writing off-planet, which is why I want to know how—without referencing known human languages like the Rosetta Stone does—it may be deciphered.
Does it help to have children's books that teach aliens to read or spell, linguistic books with diagrams of alien phonology or morphology, or a Rosetta Stone with alien languages or dialects?
All successful real-world language decipherments—Linear B, Egyptian, Hittite—have involved connections with other known languages. Linear B (Mycenaean) is closely related to Classical Greek, for example; Egyptian is related to Coptic and a bit more distantly to Hebrew and Arabic; Hittite is a distant relative to the whole Indo-European family.
Even if you only look at the writing system, figuring out Linear B involved connecting known place names like "Knossos" with words that were more common in inscriptions at Knossos than elsewhere. Ventris's biggest breakthrough came when he hypothesized that Kober's ??-C₄V₂-C₂V₂ might be k-no-so, which gave tentative values for C₂, C₄, and V₂, and extrapolated from there.
With an alien language, we have no idea what names they used for places. We don't have any loanwords or names transcribed in other languages to compare against. We don't know if they use anything resembling phonetics to communicate—or if their language is even fundamentally based on recursion like ours is.
Maybe humans and Thulians have never made contact before, but the Borean language is known, and an ancient Borean imperial decree—translated into Thulian—is found in the excavations.
Maybe Thulian and Borean are part of a language family—that hints to the linguists that Thulian might use a stack-based syntax (or whatever other weirdly inhuman feature you like) just like Borean does.
Maybe there's some old Thulian technology stored in Area 51, taken from a crashed spaceship; it contains records of an early Thulian attempt to figure out human languages (maybe even a rudimentary grammar of English written in Thulian).
Maybe they can find a live Thulian, or, failing that, an artificial intelligence of some sort, which can give them real-time feedback—a native speaker is orders of magnitude more useful than any inscriptions can ever be.
Maybe Thulian is related to some obscure Earth language(!). There used to be contact between Earth and Thule, which stopped millennia ago, and Thulian shares a distant ancestor of Etruscan/Sumerian/Pirahã/your favorite real-world language isolate. (Though be prepared for a lot of strong linguistic opinions if you decide to weigh in on the Pirahã controversy!)
Maybe universal grammar (in whatever form you like) is universal even across planets, and Thulian still has recognizable levels of phonology, morphology, and syntax.
None of these on their own would make decipherment easy—but the more you add, the easier it gets. At some point, it becomes possible, and that's what matters for a story.
I'm going to assume that the aliens' language is intentionally easy to decipher and that they are actively trying to allow other species to piece together the meaning of what they produce. I'm also going to assume that these are written or written-ish artifacts and what they look like might differ from how the aliens actually communicate in person.
I don't think the semantic content of individual signs is going to be obvious or easy to infer, but you can keep their overall number low.
One can have conventions in the orthography that make the grammatical structure really obvious.
One can have no irregular inflection and encode the parse tree directly in the orthography.
( and ( possible ( ( not have ) ( and [ Change Word ] ( not ( can predict ) ) ) ) )
( possible ( ( encode directly ) [ Tree Meaning ] ( in [ System Writing ] ) ) ) )
The example above consistently marks non-predicates with Capitalization and uses brackets to mark compounds where the meaning of the whole compound is pragmatically inferable from the meaning of its constituents but not directly implied by them.
Another idea is to make the thing that humans discover a computer or some other kind of interactive piece of alien technology.
It would be easy for humans to test their hypotheses about how the aliens' language worked if the computer was capable of answering questions formulated in the language or if the program would attempt to correct mistakes and either show the corrected response or prompt the user to pick a corrected response before answering.
There have been serious attempts at this by NASA: the Pioneer Plaque and the Voyager Golden Record. The intention is for them to be received by aliens, who will be able to use them to find out about human culture (and then hopefully won't come over to subjugate us with their superior technology!)
They use lots of universal physical and mathematical quantities as references, and with the understanding that any sufficiently advanced society will be able to identify them. For example the molecular structure of hydrogen, and a diagram of the solar system.
So one key element for easy decipherability would be to have texts available describing something universally known across the universe.
Another would be to have a simple writing system. Hangul is based on the positions of the vocal tract, which would be an aid for deciphering. It is somewhat complicated by combining multiple symbols in blocks, similarly to Egyptian hieroglyphs.
A simple grammar like toki pona would help to quickly discover regularities in the language. There are separators between the verb and the object, and there is usually only one verb per sentence. Complex sentences are broken down into a sequence of multiple sentences. Even without understanding the words, it is easy to identify the syntactic structures.
Texts could be formulaic in structure. One clue that helped decipher German messages encoded with the Enigma machine was their fixed structure, and the fact that they always ended in the same phrase, "Heil Hitler" -- that served as a useful anchor point when trying to work out the code. If your alien texts also have a similar structure, then you could use that to identify greetings or other formulaic expressions.
The most difficult aspect would be the vocabulary, especially abstract words, or polysemous words (ie words with multiple meanings). For this simple texts, or even encyclopedias and (monolingual) dictionaries would provide a good entrance point.
Most human languages follow certain quantitative regularities, which we now know about. There has been some work on the Phaistos Disc which concluded that if it is a text in an unknown language, it is most likey to use a syllabic writing system, ie the symbols stand for syllables rather than letters. This can be derived from the frequency distribution of the symbols. Of course it might not be a text -- my pet hypothesis is that it is part of a board game :)
In short, there are a number of aspects you could take into account for this. With what we know about languages and their properties nowadays, it should be quite convincing for a space mission to have a linguist as part of their team who could work on deciphering any foreign languages without too many problems. Of course it would also depend how close the alien culture was to human cultures. Language is only part of the puzzle.
It is even almost impossible for human languages. For all successful decipherings of historical writing systems some kind of clue was needed in the form of a related extant language or dictionaries and/or bilingual documents with one already known language.
When a human group wants to leave something long-term decipherable, maybe they can prepare some teaching materials with rich illustrations to convey the gist of their language. Later readers may come up with some artificial pronunciation for their script not unlike Egyptological pronunciation.
Throwing in aliens makes it even more difficult: Even when the aliens follow the procedure outlined above, will humans be able to understand the graphical representations and pictures they use? Their body parts and their home environment may be very different from ours.
"Omnilingual" was mentioned in comments, and there's your answer; assuming you have two technologically and scientifically advanced cultures, there are universal constants that can be used as a basis that historical examples on Earth didn't have, and thus we can't use to decipher old languages. The story used the discovery of the Martian periodic table as the clue that allowed the breakthrough because there is only one periodic table; chemistry and atoms are the same everywhere. To use an example inspired by that story, if you see something like "frazzlump + frazzlump + bork = blarg", and "frazzlump" is the first entry in the table and "bork" is the eighth, you know "blarg" probably means "water".
