Understanding the Apple 1 power supply
Why not take the +5V from +12V?... To begin with... look at the diodes. MR500 is rated for much more than N4001 in terms of current [3A vs 1A, if I recall correctly]. Also note the larger 5300uF cap on the 5V rail. Both indicate a higher current is needed from this rail. Also note that despite similar names, the LM323 and LM320 have different current ratings. 3A for the former but only 1.5A for the latter. Furthermore, as Passerby notes, the Stancor P-8667 is rated for 1A output while the 
P-8380 is rated for 3A.

The -5V rail probably needs less current as Peter Bennett says because of the internal construction of the Apple-1 computer, which isn't shown here. After looking a bit through the Wikipedia pages, the Apple-1 CPU (a 6502) was not actually TTL-based (as you might infer from Peter's answer), but was made in the first generation depletion-load NMOS, which indeed needed only one power supply, thus the emphasis on the 5V rail.  You can see in this low-quality schematic that the Apple-1 CPU indeed used only the 5V rail. The 6820 PIA used that +5V and the +/-12V rails as well according to what I can see there. The -5V rail was used by the RAM. Apparently this was Mostek 4K dynamic RAM, which needed +12V (VDD) and +/-5 V (VCC/VBB) supplies.

The 2400uF caps are the filter for the bridge rectifier. The input caps for the LM320 are omitted altogether because of the close proximity of the rectifier caps. The filter caps are normal in size for linear psu of this kind (high output current). You can look at Full-wave bridge rectifier with capacitor filter and ripple voltage for calculation details. To be more accurate, the +/-12 supply is split/dual rail bridge rectifier, like you usually see in audio applications. The +5V one is classic bridge rectifier.

And finally regarding the output caps (after the regulators): the datasheet shows minimal values to guarantee operation. Larger caps generally don't hurt [stability] with these old (high-dropout) regulators, but ensure a better transient response with fast variations in load, which is not uncommon for digital circuits. A much more recent ATX supply (which I've seen use LM7912 and LM7905 for the low power negative rails) also has fairly high output caps; one or two hundred uF are easily seen these days after a linear reg (because of the advanced in electrolytics that make them cheap enough) and if you look at the datasheet recommendation... it's nowhere near that.