Have you ever asked yourself, why there is a Start Button that has to be pressed to enter a game? No? Well I have! Why doesn’t the game just load up and lets you play? Why does it force you to sit through a dialog which consists of a message telling you to “Press Start”? Seems like a waste of time to me, especially nowadays where even short loading times a frowned upon. It turns out, the answer is quite interesting.
The “Press Start” screen in a video game is also called the “Splash Screen” and there are actually several reasons why they exist:

The Retailers:

Retailers and shop owners like the Splash Screen: It enables them to showcase or demo a game when it’s running on a console in a store. Think of games like FIFA. If you don’t press start for 10 seconds the game will show you a trailer. This catches the attention of customers and forces them to buy a FIFA game every year (that’s my theory, anyways).

Nostalgia:

Another reason is nostalgia. Games used to be played only in arcades, where you had to insert a coin to play it. Back then the Splash Screen said “Insert Coin(s)”. When games made their way into the living rooms, developers chose to keep the screen, as it gave the player a similar experience to the arcade.

Loading:

Splash Screens can be used to hide that the game is still loading.
When the game is started, a lot of sub-components have to be loaded. This contains the graphics engine, animation engine, sound engine and a lot of other stuff. To prevent the screen going black until everything is loaded, the graphics engine is loaded first. It can then present a (static) Splash Screen. The time the user takes to press the Start Button is consequently used to initialize all the other components needed for the game. Without the Splash Screen the device would appear to be frozen while the components load.

The Right Profile:

On modern game consoles, multiple profiles can be logged into one device by using more than one controller. When a game is started it has to know, which player’s save game it should use or to whom to award Gamer Score points. The Start Button is used to capture the profile / controller which the player is on. The 'Press Start' screen is a way to do this – the player who presses the Start Button on his controller first is considered the user.

With the current gaming console generation Microsoft, as well as Sony quietly removed the Start Button. Sony's PS4 Controller now has a "Options" button while Microsoft's X-Box One uses an unlabeled button with an icon.
Nevertheless the Start Button lives on - most developers still require the user to press a specific or any button to enter a game.