On the vast majority of banknotes, it doesn't mean much of anything anymore. Historically, banknotes were redeemable (at times practically, at other times only theoretically) for a specific amount of precious metal (usually gold). But that hasn't been true for most currency in decades. For example, the Bank of England has this to say about the phrase on their banknotes:
> The words "I promise to pay the bearer on demand the sum of five [ten/twenty/fifty] pounds" date from long ago when our notes represented deposits of gold. At that time, a member of the public could exchange one of our banknotes for gold to the same value. For example, a £5 note could be exchanged for five gold coins, called sovereigns. But the value of the pound has not been linked to gold for many years, so the meaning of the promise to pay has changed. Exchange into gold is no longer possible and Bank of England notes can only be exchanged for other Bank of England notes of the same face value. Public trust in the pound is now maintained by the operation of monetary policy, the objective of which is price stability.
Old banknotes were papers which promised that the bearer of the note could trade it at the bank for the stated value (usually in precious metal ilke silver).
If you had silver on account at a bank, instead of hauling that around, you could get a paper which said that the recipient could go and pick up the silver whenever he or she wanted. Now we get to exchange paper instead of silver.
Most currencies today are central bank notes and are not backed by a physical commodity, but sometimes old language can hang around for legal and traditional reasons. I suspect that they represent a legal fiction where they still appear. Currencies that don't have this language have different traditions.
The person making the promise is the person whose signature appears on the note -- in this case, I suspect you're talking about notes issued by the Bank of England (as that's the wording that appears on them), and the signature is that of the Chief Cashier, currently Victoria Cleland. Originally, the signature went directly below the text ("I promise to pay the bearer on demand the sum of...").
It means what it says: if you go to the Bank of England bearing a £5 note and say to the Chief Cashier, "I demand you pay me five pounds," she will give you five pounds' worth of gold.
Originally -- back in Henry VIII's time -- this would have been five pounds Tower (a "Tower pound" equalled 12 ounces) of sterling silver, hence the name "pound sterling", but in the 19th century the pound moved to the gold standard, where one pound of currency was set equal to 113 grains (about a quarter of an ounce) of 22-carat gold.
A banknote, you see, isn't an actual thing of value; it *represents* a certain amount of gold. Basically, a banknote is little more than an IOU from the issuing bank.
At least... that is, or was, the theory. In practice, if you did storm into the Bank of England with a £5 note, the most likely response (besides calling security) would be: "Sorry, we can't give you £5 worth of gold -- it's locked in a vault elsewhere and in any case we're not on the gold standard any more. But here's an IOU in the form of a new £5 note."
It's generally old text that no longer means anything. Originally, bank notes were issued by a bank to represent *real* money (e.g. gold) the bank had, but which wasn't really suitable to carry around all the time. Instead of physically giving you actual gold for a purchase, I'd give you an IOU from a bank saying "we owe whoever has this note $X in gold." But you probably didn't want to deal with physical gold either, so you could just hang on to that something and give it to someone else instead of giving them physical gold. At any point, someone could take this IOU to the bank and get it converted to real money.
Over time, these IOUs became just as much "real" money as the gold was; there was generally no reason to convert them to gold. Laws were passed making it so that they counted as a legitimate way to pay off a debt, which gave them a second thing (besides getting gold) that they were legally guaranteed to be useful for. Since exchanging them for gold no longer served much purpose and made it harder to keep the economy running smoothly, governments dropped the idea that they just represented real money. At this point, banknotes *are* real money, not a promise of a certain amount of a commodity. But some designs might still keep the old text.