I work at Google. When I was interviewing, I was in the exact same position of suspecting I shouldn't tell them my salary (which I knew was below market rate at the time). I read the same advice you did and had the same reservations about it. Here's what happened: I tried to withhold my salary information. The HR person said she had to have it for the process to move forward and asked me not to worry about it. I tried to insist. She said she totally understood where I was coming from, but the system didn't allow her flexibility on this point. I told her my salary, truthfully. I received an offer which was substantially greater than my salary and seemingly uncorrelated with it.

My optimistic reading of the situation is that Google's offer is mostly based on approximate market salary for the role, adjusted perhaps by how well you did at the interviews, your seniority, etc. (these are my guesses, I don't have any internal info on how offers are calculated by HR). Your current salary is needed due for future bookkeeping, statistics, or maybe in case it's higher than what Google is prepared to offer and they want to decide if it's worth it to up the offer a little bit. That's my theory, but keep in mind that that it's just a bunch of guesses, and also that it's a big company and policies may be different in different countries and offices.