How would I attempt to brute force a RSA encrypted message with a 64bit computer if it the memory can't put up with calculations bigger than 64bit? Without going that far to attempting to crack it, how does my computer encrypt a message with a public key that's bigger than 64bit (e.g 2048)?
Most RSA implementations are designed to be run on 64-or-fewer-bit computers, so you won't have a huge disadvantage. However, if the key was randomly generated (with at least 128 bits of entropy), then your 64bit computer will not be able to crack it anyway.
A 64bit computer can work on numbers of greater than 64bits by dividing the number into pieces. This is analogous to doing the multi-digit arithmetic operations you learned in elementary school.