SWEDISH WORD OF THE WEEK: WORD #157
All previous words can be found here!
1. modest, unassuming
Good choice. :)
språka here means to speak, and anspråka would have meant to "speak to" in the sense of addressing.
anspråk, then, came to be used for making a formal address, probably in court. Today, it means "claim" (as in the noun), and has the same history in e.g. the German Anspruch. You'd even "göra" anspråk, which mirrors the English to "make" a claim.
Hence, with -lös meaning "-less", to be anspråkslös literally means to not lay claim to anything. :)