Was it really that easy to become a knight in the time of Henry IV? Are we supposed to assume he’s a disillusioned old codger who proved himself in battle once or twice as a youngster?
Is there some back story I’m not up on? 
I don’t know if this story is apocryphal, but I have heard that Falstaff is based on a real person named Sir John Oldcastle. In fact, that was the character’s original name, until Oldcastle’s descendants objected and the name was changed.
Hey, if Elton John is a knight, why not Falstaff?
I woke up one morning and Bonzo’s keeper was President of the USA. What a long strange trip it’s been.
That is true, actually. The original Oldcastle was something of a proto-Protestant; there’s a good deal of criticism getting written about what bearing that has on the character as Shakespeare wrote him (a lot of Falstaff’s language, for instance, satirizes the Puritans).
In fact, there’s a contemporary play called Sir John Oldcastle in which the prologue makes it clear that the Oldcastle of this play isn’t the reprobate cavorting onstage at the Globe…and, as a side note, the Oxford edition edited by Gary Taylor and Stanley Wells changes Falstaff’s name back to Oldcastle! This seems wrongheaded to me, though, as the character is such an icon, and “Falstaff” such an evocative name. (To quote Harold Bloom, can you imagine a Verdi opera called Oldcastle?)
None of this answers the original question, of course; I’m not sure how much we can infer about Falstaff’s background and knighthood from the play, although we learn a little from Justice Shallow in Henry IV part 2, who in this passage is in his usual mode of reminiscing about his youth:
“There was I, and Little John Doit of Staffordshire, and black George Barnes, and Francis Pickbone, and Will Squele, a Cotswold man. You had not four such swinge-bucklers in all the Inns o’Court again. And I may say to you, we knew where the bona-robas were and had the best of them all at commandment. Then was Jack Falstaff, now Sir John, a boy, and page to Thomas Mowbray, Duke of Norfolk…The same Sir John, the very same. I see him break Scoggin’s head at the court gate, when 'a was a crack not thus high…” (3.2.19-31)
Again, not much help, but I also refer you to Falstaff’s own comment about himself:
“My lord, I was born about three of the clock in the afternoon, with a white head and something a round belly. For my voice, I have lost it with halloing and singing of anthems. To approve my youth further, I will not…” (1.2.185-189)
I think, though, that it’s kind of irrelevant to speculate too much about Falstaff’s background; he is, after all, the quintessential proponent of living in the moment – which is why it’s so weird to think about a young Falstaff! He’s kind of outside of history (in several senses) so his own backstory doesn’t seem to have much bearing on the play. IMHO anyway.
Parenthetically, a friend of mine suggests he won his knighthood in a pie-eating contest. Though when pressed further he said something very like the opinion suggested in the OP. 
This would be my first guess too. 
Yes. Nicely-Nicely Hotspur came second.