ASLR is a hide-and-seek game: in case the attacker succeeds in overflowing a buffer and overwriting pointers, the OS loads the application code (the main executable and its DLL) in randomized locations, so as to make it harder for the attacker to actually hit a meaningful location. By construction, it works better when the playground is larger. The extended address space in 64-bit mode is a larger playground.
To quantify things: in 32-bit mode, the address space is, at most, 32-bit wide (actually 31-bit when the kernel is not aware of 64-bit mode). The DLL loader must respect page alignment, and a page is 4 kB on x86. This gives, at most, 20 bits of entropy in the randomization of ASLR, but in practice lower than that because spreading the DLL throughout the whole address space can induce higher fragmentation (preventing the application from allocating, e.g., a 1 GB continuous buffer, something which is commonplace when editing pictures or movies). This is what the text you quote alludes to. Let's assume, for instance, that the DLL loading will occur only within one eighth of the address space, i.e. 29 bits; with alignment, this means 17 bits of entropy, aka 131072 possible locations for any given DLL. In any case, an industrious attacker could rely on luck, by simply repeating the attack (with a script) a few dozen thousand times until it works.
With 64-bit mode, the address space is extended to 48 bits, lowered to 44 bits on Windows, because they did not bother to go beyond. This adds 12 bits of entropy; in other words, it multiplies by 4096 the attacker's effort (in case the attackers relies on luck).
Therefore, it is true that, in some way, the extended address space of 64-bit mode increases security. Of course, ASLR has any effect only when a buffer overflow has occurred, in which case you are already in big trouble.