(Not a linguist, so take this with a grain of salt, but I believe it's accurate.)
So, first we need to divide
historical linguistics which is the bit of linguistics concerned with grouping languages into historical families from what we might call
descriptive linguistics which is more closely aligned to psychology and the cognitive sciences.
Let me say a bit about methods in historical linguistics. To create a family tree of languages you first start by gathering words with similar meanings that look obviously related:
- Latin: unum, duo, tres, quattor, quinque, sex, septem, octo, novem, decem.
- Spanish: uno, dos, tres, cuatro, cinco, seis, siete, ocho, nueve, diez.
- German: ein, zwei, drei, vier, fünf, sechs, sieben, acht, neun, zehn.
- English: one, two, three, four, five, six, seven, eight, nine, ten.
On a first glance, it's obvious that Latin and Spanish are more closely related to each other than to German and English, and it's obvious that German and English are more closely related to each other than either is to Latin or Spanish. So we now have two candidate families. But notice there are some similarities between the two families---the words for "six" sound similar in all, for instance.
So perhaps we come up with a hypothesis: there was once a language which was the
last common ancestor of the Latin/Spanish family and the German/English family. To flesh this hypothesis out I need to do two things:
- I postulate some primitive vocabulary for this last common ancestor language, and then
- I postulate some
sound change rules that would let me systematically transform the primitive vocabulary of the last common ancestor language into the vocabulary of the descendent languages.
For sake of concreteness, let's get a proposal on the table. (Not being a linguist, I'm sure the actual claims are wrong, but it's just an illustration). Suppose I call my last common ancestor language
proto-indo-european, and I hypothesize that the word for "three" in PIE was "tre". Further, I hypothesize that the Latin/Spanish family didn't like words that ended with a soft e sound, and so they tended to add an "s" to such words, and so you get "tre" > "tres".
Suppose, however, that the German/English family didn't have a problem with words ending in a soft vowel, but then later they began to not like the initial consonant cluster "tr". Perhaps the German speakers tended to mutate "tr" to "dr" and the english speakers tended to mutate "tr" to "thr". Then you'd get intermediate forms like "dre" in German and "thre" in English. Postulating further vowel changes would give us "drei" and "three" respectively.
Of course, what I've drawn here is a really simplistic picture. Languages are constantly mutating and evolving, and they always come in a variety of dialects. The point is that each of the hypotheses above is testable.
If my historical hypothesis is true, I should expect to find some dialects of German, or some historical inscriptions in German where the word for "three" is "dre" rather than "drei". Further, we can also look at historical and archaeological evidence for examples of language change.
Also, notice that we can draw on some important non-linguistic evidence to help us chart the chronologies too---we already know from history that Latin is the older language, and Spanish descended from it and that will help us figure out that PIE is going to be a lot more like Latin than Spanish probably.
tl;dr. Historical linguistics uses a variety of methods: comparison of dialects, archaeology and anthropology to help verify its reconstructed languages.