4:15a that a search should be made of the record books of your fathers.

⌊We (excl.) suggest/recommend that the king/chief⌋ searches the record books of/from previous kings/chiefs.
-or-
⌊We want you to know about it⌋ in order that you (sing.) command/tell somebody to look in the books where scribes wrote the events that happened in the time of kings/rulers who ruled before you.

4:15b In these books you will discover and verify that the city is a rebellious city,

From these record books, he will know that this is a city that often rebels ⌊against kings⌋.
-or-
You (sing.) will find proof in these books that ⌊the people of⌋ the city of Jerusalem have always rebelled ⌊against kings/rulers⌋.

4:15c harmful to kings and provinces, inciting sedition from ancient times.

It has caused/brought harm to kings/chiefs and their territories. From ancient/past times, it has been a place where the people have often rebelled/revolted against their kings/chiefs.
-or-
⌊You (sing.) will find that⌋ many times in their past, they have stopped obeying their kings/rulers. So they cause trouble for their kings/rulers and create unrest/disorder in the provinces/regions ⌊where they live⌋.

4:15d That is why this city was destroyed.

That is why this city was ruined.
-or-
That is ⌊the reason⌋ why one ⌊of your (sing.) predecessors⌋ destroyed Jerusalem city.

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