Apparently the passage may be found in three books by Eusebius. Here is the most common.
The context of mentioning Christ is in relation to his mentioning John the Baptist and elsewhere perhaps James the Just who is brother of Christ. These things are mentioned as explanations for subsequent calamities.
It was on her account also that he slew John, and waged war with Aretas, because of the disgrace inflicted on the daughter of the latter. Josephus relates that in this war, when they came to battle, Herod’s entire army was destroyed, and that he suffered this calamity on account of his crime against John.
The same Josephus confesses in this account that John the Baptist was an exceedingly righteous man, and thus agrees with the things written of him in the Gospels. He records also that Herod lost his kingdom on account of the same Herodias, and that he was driven into banishment with her, and condemned to live at Vienne in Gaul
… 7. After relating these things concerning John, he makes mention of our Saviour in the same work, …
TO ADD/EDIT: In my opinion, this answers the OP question. What is the context for Eusebius quoting Josephus? The answer is to show why the Jews experienced subsequent calamities, including the destruction of their temple and Jerusalem. Jesus is the Christ. John the Baptist was a righteous man. They, along with Pilate and others, put them to death. It was wrong, and God poured out His wrath. That's the point for Eusebius.
- Since an historian [Josephus], who is one of the Hebrews themselves, has recorded in his work these things concerning John the Baptist and our Saviour, what excuse is there left for not convicting them of being destitute of all shame, who have forged the acts against them? But let this suffice here.
As to what seems to be another question/answer, is Eusebius verifying that Jesus is the Christ, the answer is not really. It is a fact, but not the point. Again the context for Eusebius is to explain why it (destruction of temple and Jerusalem) happened.
To be sure, subsequent Christians have used Josephus' comments as "outside" proof that Jesus is the Christ, and use Eusebius in support of that assertion, but again, that was not Eusebius' primary point.
With this in mind, I'd further argue that the quote from Josephus is in fact true with no, or very minor, subsequent interpolations. If it wasn't true as one explanation for 70ad, why else would Eusebius mention it within context of trying to explain 70ad?