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While booking hotels in the US (Dallas, TX specifically) I found that many hotels have an area that features shelves with food, fridges and freezers. I see no price tags with the products. Does this mean the products are free for hotel guests?

An example can be found here:

If these products are not for free, how do I find out how much they are? Is it considered rude to put something back if I think the price is too high?

I often stay at Radisson hotels. And have also seen this type of thing in other hotels as well.

This area in hotels is a shop featuring paid-for items.

The items are likely to have small price stickers on, however if they don't there should be a poster displaying the prices.

It would not be rude to look at the prices and return them to the shelf if you did not wish to purchase.

Pricing in these areas/shops is usually higher than you would pay in a supermarket or convenience store, but likely to be cheaper than the minibar in the hotel room if there is one. .

This looks like a feature called The Market at Courtyard by Marriott, though most business oriented properties feature something like this.

The items are not free but you can pay cash/credit or charge to you room.

I recall the price always displayed either on the shelf or sticker. It's like any convenience store. If the prices is not suitable, place the item back on the shelf.

Despite this display, the staff will be happy to direct you to a nearby store.

I have been several times in Dallas and those shelves are pretty common in Marriott and Starwood hotels. The first reason why you see those is because a lot of them are business type of hotels. So you mostly have solo travelers and they prefer to eat something quick in their room rather than taking the car to go out to a restaurant. Usually, those hotels don't have a restaurant in house, they only offer breakfast.

Then, it is always paying. The exception is the club lounge in some of those hotels where you can have "free" food but this is available only for higher room categories or loyal customers.

The price of the items available in the shelves is usually displayed upfront or on small stickers directly on the items. As mentionned, you can book it directly to your room if you don't have money. The good thing is that it goes far beyond food, you can also find a lot of travel items that can help making your travel better, especially if you have forgotten something at home.

Usually if you find such a room, it should have a sign explaining how it works. If it doesn't specify free, then it's not free. One notable I stayed at, Candlewood Suites, had an alternative method for providing snacks. The items in the room were not particularly under watch, but a sign posted in the room stated that the room worked on the honor system; if you took out some products, you were expected to leave some money in return. Some products were also free, like the brewed coffee and/or tea. Other hotels may actually operate this way as well, but you should make no assumptions. If you stay at a hotel that has a room that looks like this, you should ask in advance how it works, should you decide to satisfy a snack urge. The staff will be happy to explain how much it will cost you, methods of payment, and so on.