Hi,
Normally a PICaxe pin can't deliver enough current (nor maybe resist sufficient voltage) to drive a "buzzer", so the transistor is used as an "amplifier", which is a slight misnomer because it can only control or switch an external current, rather as a relay does. The sounder is actually the one component (in addition to the regulator) which might be connected directly to the 9 volt supply, to make it louder (because the transistor can safely resist that voltage), but that would depend on the buzzer's specification/rating. Also it's important that it is an active "buzzer" and not a (passive) "loudspeaker", which would need to be handled differently.
My comment about the layout was that the LEDs appear to overlap (they do seem quite large) and there are 8 components south the PICaxe and not much more than "white space" north of it. The white space might be a better location for the programming socket (nearer to Legs 2 and 13). If the LEDs are "off board" then they might be shown as a "header socket" to connect the wires, or perhaps look at how Rev Ed "anchor" the battery leads on most of their prototype boards.
As for the overall layout, it would normally be better to place the PICaxe nearer to the centre of the board, perhaps oriented N-S rather than E-W. There is considerable freedom in how the pins can be allocated, but it's sensible to start with the inputs (switches, etc.) on the LH Side (Legs 2 - 7), outputs on the RHS (Legs 8 - 13) and the programming socket at the top. There are advantages in taking push-switches to ground (rather than the supply) because there is less risk of accidentally damaging the PICaxe, and the internal "Weak Pullup" resistors can replace external ones. But it's probably worthwhile to include "current limiting" resistors in series with (all) the signal pins, and then they can be used also to "jump over" PCB tracks where required.
Cheers, Alan.