This is using
’s as a contraction for “has”, rather than for “is”. Thus, it is saying that it has informed Sara Bareilles.
How the sentence should be read, omitting the contraction:
So it has really informed how I am in the world in a big way.
The usage meaning “to influence” is the most appropriate definition here.
I can tell this because “informed” is not an adjective that would work here; that would mean that “it” is knowledgeable, and with the rest of the sentence it would be incorrect if read that way. Instead, it is using the present perfect tense—“has” and then a past participle, in this case “informed”.
Cool info coming from Anton Sherwood in the comments:
This usage comes from Latin “informar”, to shape/form/instruct/educate that in turn came from “in forma”, literally “into shape/form” (etymology details from Wiktionary).