Does this mean that Democrats voted in the Republican primary, so that they could defeat Laffey, then changed their affiliation after the primary so they could go back to voting Democrat in the General Election?
No, it means independents voted in the R primary. Independents outnumber Rs or Ds in RI, and they like Chafee.
Chafee probably will win the general. I don't see what that gains us, unless the Senate is split 50/50, (and even then, maybe not) but he probably will.