When you sell a copy of the software without a license agreement, traditional sale rights happen. It is no different from selling a copy of a book you wrote, selling a copy of a photo you took, or selling a pie you baked.
In most places of the world that matter that traditional sale includes an implied warranty. That includes things like "fitness for a particular purpose", "merchantability", and "workmanlike quality", among others. These implied warranties mean that your product meets certain standards and that you can be held liable if they don't meet those standards.
An EULA has two purposes.
First, it is designed to remove as many of those implied warranties as legally possible.
Second, an EULA is often designed to (attempt) to convert the transaction from the laws of a sale over to the laws of a contract. Contract law generally offers more friendly terms to the seller.
Imagine a scenario: Somebody downloads a game to their tablet. They also have data from their work on the tablet. Your game has a bug that causes the entire tablet (including their work data) to be erased.
Do I need an EULA for my own protection?
Without an EULA in place, the individual or the company could sue you under the implied merchantability warranty, and then "helpfully" offer that you can settle the dispute by paying a few thousand dollars for data recovery services. You could choose to go to court where they may not win, but the lawsuit is going to cost you a lot; it will then be cheaper to pay the few thousand in data recovery than the lawsuit.
It is all about risk analysis and risk management.
You need to understand the risk of the lawsuit, understand the costs of a lawsuit, and decide if the license is worth it. Also note that the cost to include license terms is practically nothing in the long term.
If it helps you in your risk analysis, note that nearly every single program out there includes such a license that disclaims implied warranties. Everything from the GNU GPL to the most draconian licensing terms include them.
I think most Android games, even professional ones, do not require you to read an EULA before using them. Why is this?
Many games will include lines buried in their app's store description, or through links at the bottom. They don't require an explicit agreement because of changes in what is expected. These days it is a reasonable expectation that every piece of software has a license agreement, and that agreement is mentioned in the details of the product.
A few examples of that from the Google Play store: Angry Birds, Photoshop Touch, Plants Vs. Zombies, and just about anything else you can think of.
Most people don't take the time to read the details or fine print on the app store, but simply having the fine print included on the site is generally enough to satisfy the lawyers.