Powering a Digital Alarm Clock with AC
With that circuit the AC is absolutely 100% required.  There is no getting away from it (short of generating your own AC signal).

The AC:


Provides all the timing for the clock.
Provides the duplexing for the display.
Provides the main power through a very crude rectification system.
Provides the "buzz" of the alarm.


The "50/60Hz" input is used as a clock input, which is divided by either 50 or 60 to create a 1 second pulse against which the clock's timing is counted.  A slight variation in that input frequency will mean clock drift over time.

The display output is multiplexed - it does hours in one half of the AC cycle, and then minutes in the next.  As the AC wave changes the diodes connected to the transformer alternate as to which are sinking the current from the display.

If you can generate an exactly 50Hz or 60Hz signal with enough precision to not cause the clock to drift (or it would be a bit useless), and that signal can sink enough current to handle the display (100mA or so), then you may be able to get away without the AC transformer.

As for why your caps blew... well, an 11.8V transformer, after rectification and smoothing, would be nearer 16.7V.  Best to use a smaller transformer, like 7-9V.