What Satoshi is referring to is the traditional financial system's inability to transfer small amounts of money. Traditional money transfer systems such as ACH and various wire transfer services are reversible for a certain period of time, meaning that there is some risk involved. The risk that the bank will have to mediate a reversal in payment (which costs them money) makes it so that they need to charge fees in order to transfer money on your behalf. If the fee is not worth the risk, the bank won't do the transfer.
Let's take the example of paying a friend $5 for picking up the tab at lunch. You tell your bank to transfer $5 to his bank over the ACH system. This system takes 3-5 days to complete the transfer. During that time period, a reversal is possible, and if it occurs, the bank needs to use its resources to mediate it. This might include retrying the transaction, contacting the other bank, or contacting the customer in the case that the $5 has already been withdrawn. These actions all cost the bank money. If the probability that a reversal will occur,
P, multiplied by the expected cost to the bank for dealing with the reversal,
C, is greater than the fee the bank will earn,
F, then the bank shouldn't allow the transaction. So we have a system where the following must be true:
F > C x P
Let's say the bank charges a percentage as a fee, let's say an outrageous 10% of the transaction. That means 'F' is $0.50. In this scenario, it is unlikely that
C x P will be less than $0.50, so the bank can't do the transfer.
Let's say that the bank charges a flat rate for transfers, one that will cover any of the risk. A value of $10 for a transfer of any amount would not be unreasonable from the bank's standpoint. However, would you be willing to pay the bank $10 to transfer $5 to your friend?
This is just one of the many problems that Satoshi solved with the invention of the Bitcoin blockchain.