What was Big Willy Shakes going for?
Shakespeare coined the phrase "send him packing," which appears in print for the very first time in this scene of Henry IV Part 1. It simply means Falstaff will tell the guy to scram, vamoose, get lost, beat it, etc.
Falstaff is standing up for his pal Prince Hal here because he doesn't want him to run off to the king. And Hal sure doesn't mind. Hal is hell-bent on rebelling against his father. For Hal (and audiences) Falstaff is the embodiment of rebellion and disorder. Critics often link his character to "carnival," a religious festival season that celebrated unruly and riotous behavior, much like Mardi Gras today. But carnival was temporary, and Falstaff lives this life of raucous behavior all the time over at the Boar's Head Tavern.
Falstaff takes bribes from able-bodied soldiers and recruits a rag-tag crew that's not fit for battle. He loves to party and always has a clever, good-natured joke at the ready. He lives the life that Hal wants but knows he can't really have because he is a royal (and the heir to the throne at that).
So it's no surprise that Hal's dad—ahem, the King—doesn't want him hanging around Falstaff. And it's no coincidence that when the messenger from Hal's dad shows up, it's Falstaff who eagerly volunteers to send him away. If Hal is obedient to his dad, then he can't be friends with Falstaff. So Falstaff is more than happy to send that dude packing.