I asked a friend who does certification for console games for a major studio (certification is the process to get it approved to be released on the console by the vendor).
He said there's a requirement that the game must have some interaction with the user after a set time period, even if the game isn't fully loaded yet.
The "Press Start" or what
have you is to meet that requirement: the game only has to load that far within the time limit then the user can say when they're ready to load the rest of it.
There's also some conventions involved.
Some games if left to sit on that screen have a video they can show, and some games will simply wait for input to figure out which controller is the one to let set things up (though other ones just require controller 1 to do that).
gaming.stackexchange.com/users/7996/Tridus
Consoles typically have more than one controller, so they likely require a button press from a player to determine which controller that player is using.
This way, no matter which controller each player is using, after the game loads you can determine who to consider 'Player One' by asking them to press a button.
PC does not need to do this because there is only one player and one controller: the mouse/keyboard.
Edit:
This was a guess, but there are actual game certification reasons for having the extra button press.
Please read the other provided answers.
gaming.stackexchange.com/users/1721/Sadly Not
The true reason is because it's a requirement that the console manufacturers enforce if you want to create games for their systems.
PC games don't have requirements like this.
There are some convenient side effects of the Press Start screen, though, which probably help explain why the requirements exist:
When a user presses start, the game knows who's "in control" of the game at the moment.
This means the game can do nice things like display the game environment in appropriate context.
For example, if you have a saved character, the game can show that character.
It's a convenient place to put branding info, like company logos and such, as well as whatever legal text might be necessary or stuff like ESRB info.
It's a good landing page for the (also required) "
Attract Mode" that shows something interesting every so often like a movie or some gameplay.
The game can load up the Press Start screen while other stuff is loading in the background.
It gives the player something to look at if the intro movies are over but the rest of the game's front end isn't ready to go.
You might notice that some Press Start screens actually don't show the "Press Start" text until several seconds after the screen appears (and if you noticed that, you're pretty dang observant, good job!).
If we didn't need a Press Start screen, I think we'd just skip making it most of the time (unless we wanted some of the benefits above I guess).
Note also that downloadable games may have different requirements than full retail games, so this might not be consistent across all the console games you've played lately.
gaming.stackexchange.com/users/32/Mag Roader
Another game certification reason that hasn't been touched on above is cert requirements (TCR, TRC, etc.)
usually require some kind of 'attract mode'.
That is the game should do something like replay game footage, show an intro video or something else at least moderately interesting if you just turn it on and leave it.
Attract mode exists so that when the game is loaded up and just sitting there at a Gamestop, convention booth or other installation it provides a little visual flare to attract attention.
The XBLA game Braid is notable here because it doesn't have an attract mode.
Braid's main screen acts as the level select screen and you control the character directly on it.
gaming.stackexchange.com/users/7999/Michael Apfelbeck
My PSP does this as does my XBOX, thus I'm sure it not about which controller is in use.
I always assumed it was a licensing issue, perhaps to ensure the user as read the copyright notices and such.
Why non such requirement on a PC?
I'm guessing that's because PCs are not closed systems.
Anyone can write a program for a PC, but to write one for an XBOX, PS3 or PSP requires special licenses to be signed.
I'm not sure where this sits with community games on the XBOX360.
gaming.stackexchange.com/users/7991/dave
I haven't seen anybody answer this
but I always thought that it was because that way game stores could just leave it on and the games would loop playing some type of video that showed off the game and then return to the "Press start" screen.
gaming.stackexchange.com/users/1427/Juan Besa
I would add that older TV's may have had burn-in issues if the same menu screen was displayed for a long period of time, while the computer would have a screen-saver kick in instead
gaming.stackexchange.com/users/83202/user2813274
I am strictly a PC gamer myself, but I noticed a strange thing. When you start every released-for-console game, like Mass Effect or Borderlands, they ask you to press Enter (or any key, etc.) after the initial load, and only after pressing it do you get to see the main menu. PC-only games load the main menu immediately.   What's the reason behind this? I assume it must have something to do with the way consoles work, but what purpose does this extra button-press serve?
Why do console games require a button press before showing the main menu?
gaming.stackexchange.com
<xbox-360><ps3>
gaming.stackexchange.com/questions/18780
gaming.stackexchange.com/users/7207/Sejanus
The gamer must have some interaction with the user after a set time period. Some games have a video they show if left untouched for a long time. It gives the player something to look at if the game isn't ready yet. Games are using different techniques to tackle user innactivity and technical problems. The press start screen is such one and provoking for an action is a way to deal with it.
It is suggested that there are specific reasons why such an action is required from the user, such as the detection of a controller.