My understanding of the shift in meaning is that the usage of the word "import" meaning "to bring in" became synonymous with "of significance". And "importance" eventually became the more accepted form (perhaps because of differentiation.)
For example: in the Jane Austin Novel "Pride and Prejudice" a character is mentioned to have a question of "great doctrinal import." At the time of the writing of the novel (1813) the word "import" (aside from the bringing in of goods) meant "To bear or convey" or "To be of weight to" (see Webster's 1828 dictionary -- as the second and third definitions state.) So even at the time of the novel, the question could have been reworded as a question that "great doctrines bear" or "of bearing to great doctrines" or perhaps earlier usage would have been "brings in great doctrines" or "brings great doctrines into question."
SIDE NOTE: here's an interesting example of shift in usage of "import" to "importance" being used interchangeably in "The Catholic World" on page 744, published in 1887.
Now I'm not saying that the English usage is what changed the meaning but I'm using the examples and definitions above to show the similarity in usage and meaning. In 1800's upper English society, speech and thought across English, French, Latin, Spanish, etc., were very tightly knit, so it makes sense that the fluidity of the usage for the word "import" slowly lent it to move from "something brought in" to "something brought in with the implication of significance" to finally just "of significance."
This is one of the natural language changes that is hard to trace, but I hope to have illustrated the similarity in meaning enough to show how it could have naturally changed just by the usage. Whether it changed in the root language or later, the usage implying significance would have naturally changed the meaning over time.