If the casting of a spell is still in progress on the server, you'll hear the familiar snapping sound when you mash buttons. If casting is completed on the server, but your computer doesn't know it yet, it will send a new cast command anyway, thus initiating a new spell.
An important side-effect of this change is that button-mashing for spells with cast times longer than 1.5 seconds will be penalized more. This is because every time you punch that button, the global cooldown will be triggered every time. It will be 1.5 seconds later (or earlier if the server interrupts it in the event of a spellcast in progress) before your button presses will have any effect. This means little in casual play or grinding, but will have definite repercussions in time-sensitive scenarios such as Arenas, or some raids.
Currently, spells with long cast times can be spammed more effectively with the /stopcasting command and the Quartz addon. For example, my warlock uses the following macro to maximize shadow bolt dps:
/cast Shadow Bolt
With the Quartz latency module, this macro can apparently increase dps by 5-20%, especially for players with high latency. This 'trick' overcomes the lag between you and the game server and allows your computer to send a new cast command, without waiting for the server's acknowledgement that the spell is done.
What does this change in 2.3 mean for players? It means that you can dump the macros for chain casting and go back to using normal spell buttons (those that you drag out of your spell book), as there is no need for the /stopcasting command for this purpose anymore.
When patch 2.3 goes live, the /stopcasting command is also not needed for macros that combine multiple abilities and spells.
For a more detailed, and more technical, explanation, check out Altitis' blog. Quartz is available here.
Every button press of an ability that triggers the global cooldown (GCD) will trigger the cooldown come 2.3. However this doesn't mean that four quick button presses will net you six seconds of GCD. The server will check if an ability is in the progress of casting, and cancel the subsequent GCDs if that condition is true. If no casting is in progress on the server, GCD kicks in on your computer and it will be 1.5 seconds before you can cast anything else.
Effectively, the check for valid casts (outside of GCD) has moved from client-side to server-side.