Why would a seller at an auction set their reserve price hidden?

"Do not let the other side know how much it will take for him to get the goods out of your hands"

If we were to write "The Manual of a Good Negotiator", the previous phrase could easily be at the title of one of the top three chapters in that book.

These are some other rules if you want to turn yourself into a good negotiator:

1. Play fair to you and the other party
2. Never let the other know how much you paid for the goods you are offering
3. Talk about benefits, not so much about price
4. Never say that you need to sell
5. Let the other person name a price first. If you must name a price, name one far higher than the one you are willing to negotiate at
6. Reduce the price gradually, not at once and always pushing the other person to offer more. Talking about the benefits provided by the goods being sold gets you closer to a deal than anything else
7. Let the other party increase the price until he arrives at your desired price. Good negotiators are the ones who say "Deal", not the ones who offer to pay more or to sell cheaper.
8. If you are buying, make sure that you run every meaningful inspection related to the goods or properties you intend to buy, and use the red flags on those inspections to reduce the price to a point lower than "Price minus restoration costs", this is because your restoration endeavor deserves compensation

I found this puzzling also at first. But after reviewing many past successful and unsuccessful ebay auctions, I came to this conclusion: Keeping the reserve price a secret is all about the psychology of people.

Many people are hesitant to be the first bidder. But, if they can bid very low, they are more likely to bid. Some are afraid that they will be a sucker (paid too much). A second person sees that someone else is interested and puts in another low bid. Then a third person, etc. If the reserve is reasonable, it will be reached reasonably quickly.

When people see a lot of other people bidding, they get caught up in the bidding frenzy. When a reserve is known, often nobody will put in the first bid. This happens even when the reserve is lower than the price at which similar items usually sell.

I agree with the other answer about psychology, by starting at a price of $1; you're definitely below market and may attract an unusual amount of bidders that way which could lead to a bidding frenzy.

Another angle is price discovery of rare, or rarely transacted, items. It's hard to figure out at what price something rare should transact. When you combine a lack of comparable transactions with an unmotivated seller you can get to potentially a way above reasonable reserve price and an item that doesn't sell. The next time item hits an auction, the reserve is more likely to be public and somewhere in the range of the ending of the prior auction.

To the point you allude to in your question, a private reserve price for an Ebay auction of a commodity item like an iPhone is more or less pointless or indicates a seller attempting to capitalize on the psychology effect outlined in the other answer.