It is found in an exhibition of Qin Dynasty held in Xi'an city. The exact words are 秦国35代国君发奋图强，做好了“大出天下”的准备。I have checked the words online for the meaning, but finding no answer.But roughly I think it might mean something ambitious, because it is about a growing country with some hard working kings.
Let's look at similar phrasing words
大赦天下 grant amnesty within China on a grand scale
大出天下 sounded like "on a grand scale, set off from Qin to conquer the rest of China".
Since Qin was preparing to unify China around that period, '出' in '大出天下' was more likely suggesting "出咸谷關" (out of Qin's border) or '出征' (send off the army/ campaign)
It's good to think in classical Chinese.
大is adverbial on
出emphasising its extent. Similar constructions include
大叫(to yell loudly),
大敗楚軍(to defeat utterly the armies of Chu) etc.
大出天下feels like neo-classical Chinese to me (which is a very effective means of communication in the context of a historical exhibition). It's not an established expression in the classics, but understandable. With the Qin rulers being the grammatical subject of the sentence,
出perhaps can be parsed very literally as in exiting the borders of Qin, but this clearly conflicts with the grammatical object
天下('all that is under heaven'): it makes no sense to exit the target of your conquest.
出here is similar to that in
水落石出(stones appear as water recedes). For understanding, I think it does no harm to make clear the intransitivity of
出(meaning 'to appear' instead of 'to exit (sth.)', see eighth definition here) by adding the preposition
天下to make the phrase adverbial, as in
大出天下 ↔ 大出於天下 (phrase adverbialised) ↔ 於天下大出 (phrase inverted)
to make Qin known in all that is under heaven
Similarly, we observe the adverbial nature of
天下in the much more common expression
稱霸天下too: it must not be the object because the transitive
稱(‘to declare (oneself)’) is already matched with the noun
秦国35代 should be near the end of Qin Dynasty. At then, internally it was corrupted, externally it was surrounded by fierce enemies. Its 35th emperor, a wise, ambitious emperor decided to lead his kingdom to regain its past glory and "raise" high above (大出) the competitors/enemies who had grabbed land from its original territory (天下) and threatened its survival.
出 has the following meanings:
"to rise", "to go out", "to come out", "to produce", "to go beyond", "to put forth".