There a couple of things that allow him to make these claims without being "laughed out of the room," as you say, but they don't have to do with his personality in the sense in which I take your meaning.

First, there is the rocket equation. That little gem tells us the amount of fuel it takes to achieve a given change in velocity for a given energy density of fuel. Then there are some results from orbital mechanics and aerodynamics that tell us what that change in velocity is. For the curious, escape velocity is just sqrt(2) times the circular speed at any given altitude.

I've laid this out before, but here it goes again. To put a pound of anything into orbit has a fuel cost of a little over $20. "Incredible!," you say, "It costs $10k/lb on the Space Shuttle! How can that be?!" Like so. Typical mass fractions are on the order of 2%. That is, 2% of the stuff on the pad, fuel, structure, payload, everything, actually ends up in orbit. About 12% of that mass is structure, things like tanks and engines and the like. That leaves 86% of the thing as fuel. 86:2 is 43:1. 43 lbs of fuel for every pound of payload. Assuming that propellant is roughly as dense as water and roughly the price of milk, both easily verified, that's under 6 gallons of propellant for every pound of payload, which will run you $21 at $3.50/gallon.

Multiply sqrt(2) by $21/lb and you have something like $30/lb. If you and your capsule weigh 2,000 lbs, That's $60,000 for a one way ticket. A little over 8 times that price may be a reasonable number. So, what makes up the difference in cost for current launch systems, or even for antiquated and clunky systems like the Shuttle? Low safety margins and their concomitant need for enormous administrative costs for each part, disposable launch systems where that administration cost burns up in the atmosphere or splashes down in the Pacific, and enormous system complexity driven by a lowest-flight-weight-results-in-the-cheapest-vehicle mentality.

We can begin to address, based on SpaceX's design philosophy and planned vehicle, how they may be able to make these claims without deserving to be "laughed out of the room."

First, SpaceX has reduced engineering and integration costs by reusing common components and simplifying designs at every step. they were (and I think, still are) using a pintle injector which is much less susceptible to catastrophic combustion oscillations than the more typical injector-face solution, at a cost of some performance. The tanks for all of their stages are the same diameter, allowing them to engineer and build one capital-intensive jig rather than two or three, and they get more experience with that hardware since all their work is done on it. They're using a pneumatic stage sep mechanism rather than a pyrotechnic one to eliminate material-handling, static, and other safery concerns related to pyrotechnics. Rather than relying on one or a few very large engines to power the first stage, they've chosen to use 9 smaller engines on the first stage and isolate each one in its own cato-proof container, again allowing them to gain more experience with a single system, prove its reliability, and leverage that experience and track record to perform a larger job.

Second, they have plans for full reusability of the launch system based on incremental changes to their existing systems. Yes, there is a fuel and performance penalty for going this route, but the savvy armchair aerospace critic will note that those penalties are expressed in tens of dollars per pound, whereas 100% disposal is measured in thousands to tens of thousands of dollars per pound. That is to say, even if reusability results in a 10-fold increase in fuel cost but allows vehicle cost to be amortized to negligibility, we're still approaching Musk's $500k/flight number. As to his actual plan, the fuel cost to land a booster segment is tiny compared to the cost of launching a vehicle. The first stage will simply reenter without having to retro-burn, and the second stage will need to retro-burn just enough to enter the atmosphere to achieve the rest of the braking. After that, the delta-v required is on the order of 100m/s, hardkly the 10km/s needed for orbit. You seem to know what you're doing, so I leave the derivation of that penalty, using the rocket equation, to you.

tl;dr: You're absolutely wrong in the most irrelevant way, and had you addressed SpaceX's achievements and plans in anything like a rigorous way, you could easily have answered your own question.

EDIT: The fuel cost for escape will not be sqrt(2) times the cost for circular speed. The real factor will be something more like 2 or 3, not 1.414... Still, we're in the range of $60/lb, not $6,000/lb.