So, fundamentally you're asking about how we can be sure of the methodology we've used to reconstruct Proto-Indo-European: the comparative method. The comparative method is a process of assembling extensive wordlists covering words with similar meanings in a variety of languages that you think are related and identifying regular correspondences between them. It's crucially important that the correspondences are regular, i.e. rule-governed. Either a correspondence always holds, or it holds under a limited set of circumstances that can be cleanly described. Let's take a mini-example of the correspondence between initial voiced stops (b d g) in Latin and initial voiceless stops (p t k) in English.

English	Latin	Latin gloss
peg	baculum	'staff'
tame	domo	'I tame'
tow	duco	'I lead'
timber	domus	'home'
kin	genus	'kind'
corn	granum	'grain'

So, these words are semantically related, and a commonality we see in all of them is that where English has a voiceless stop, Latin has a voiced one. This holds true for hundreds of vocabulary items you can match up between English and Latin. Historical linguists have spent over 200 years now finding these kinds of regular correspondences between Indo-European languages and language families--some of these correspondences, like Latin s and Albanian gj (like the g in George for many speakers these days) seem improbable at first glance, but the correspondences are undeniably there and they are undeniably regular. We're talking thousands of pairs of words in hundreds of languages, at this point.

If you'd like to learn more, you might try looking for a copy of Benjamin Fortson's introduction to Indo-European. It's dense but it does an excellent job of laying out the evidence for PIE in a very methodical way.

You might be wondering how we can be certain it's not all just a bunch of circular reasoning. I've written about that here.

Your question is worded ambiguously. Do you mean:

How do we know that modern [Indo-European] languages all have one common ancestor in the Indo-European language?

Otherwise if you mean every language in the world, yours is the first mention I've ever seen of the idea.

Can you clarify?

[Note: this is a shallow explanation.]

Not all modern languages come from Proto-Indo-European (PIE). Turkish, Guarani, Basque, Arabic, Japanese, Mandarin and many, many others modern languages have nothing to do with it. On the other hand plenty European and Indian languages do: English, Italian, Greek, Persian, Hindi, Kurdish...

Europeans always noticed similarities between Latin and Greek, but they could be brushed off as two languages "mixing" with each other for a long time. Until they learned about Sanskrit - a language that is clearly similar to the two above, but not only it is in the "wrong" place (India) but at the "wrong" time (Sanskrit is considerably older than Latin). There was no way Sanskrit mixed into Latin or vice-versa.

The next hypothesis was that Latin and Greek somehow came from Sanskrit. While this, people were understanding better how languages change, /u/rusoved described the process perfectly. And they begun to apply the same process to their hypothesis ("Latin came from Sanskrit")... but there's something really wrong - they couldn't explain certain stuff, like «if "father" is "pita" in Sanskrit and "pater" in Latin, where did Latin get that "r" from?».

The next logical step then was that saying: "Latin wasn't born from Sanskrit. Both languages were born from the same, unknown language". This unknown language is what we call now Proto-Indo-European.

Solving the connection between Greek, Latin and Sanskrit, people begun to test other languages as possible children of PIE:

Oh look, German fits in just fine. So English and Swedish too, since we know they're related to German.

Russian fits in just fine. Polish, Ukrainian, Czech and others are related too, so they're IE languages too.

Finnish doesn't fit... it wasn't born from Proto-Indo-European.

Irish fits in, as well those Celtic inscriptions from Europe. Celtic languages are Indo-European.

Basque? Nah... it doesn't fit. Turkish doesn't either.

Et cetera.