I'm assuming that the "we" in your question means "English speakers". (ETA: it actually seems to mean "speakers of modern European languages", but the difference doesn't substantially affect my arguments.) If so, the claim you're presenting is partly true, but also partly false, and the part that is true is probably not particularly meaningful or important.
Here's the true part: on the whole, English does use more words to convey a given idea than either Latin or Ancient Greek. This fact is familiar to most people who've studied these languages, and is easily seen in e.g. the volumes of the Loeb Classical Library, which present Greek or Latin texts with facing English translations. The right-hand (English) page usually contains more text than the left-hand (Greek or Latin) page.
The main reason for this is that Latin and Greek are highly inflected languages, which lets them pack more items of meaning into each word. For example, a verb in these languages will include information about person, number, tense, voice, and mood, much of which would in English be parceled out into separate words: e.g. to reflect the single Latin word amabar, English needs a longer phrase such as I was being loved.
The part of your claim that's probably wrong, though I haven't tried to check this empirically, is about the difference between Latin and Greek. As someone who teaches both languages, my sense is that Latin is more compact than Greek rather than the other way around. There are a few reasons for this, such as the fact that Latin lacks definite articles, which are very common in Greek; and that Greek has a large array of discourse particles that add various nuances of meaning, which it uses more freely than Latin.
But the larger question is, given that Latin and Greek really are more concise than English, does that tell us anything important? To this the answer is almost certainly no. First, the difference probably does not represent any kind of general trend over time from higher to lower concision. English happens to be a less-inflected language, but there are many languages spoken today which are just as heavily inflected as Greek or Latin and presumably comparably concise; while on the other hand, of the thousands of languages that were spoken two thousand years ago, there were certainly many that were of the English type rather than the Greek and Latin type. So there's nothing to be concluded from these particular data points.
Finally, the subtext of your question seems to be the idea that a lower word-to-idea ratio in a language is somehow better or more efficient. But this too is difficult to argue for. First, even if we accept that concision is a virtue, why count words rather than some other unit? Why not phonemes, syllables, or morphemes? If it turned out that Greek uses fewer words than English but more syllables to express the same idea, which language is more concise? But more importantly, there's really no particular reason to value concision. The expense of producing words (or syllables, or whatever) is virtually nil, so it's hard to see why the words-per-idea ratio should matter in any way. The extraordinary success of English as a global language suggests that it doesn't.