It gives you time for genuine discovery. Burn out, if you've ever experienced it, is not about exhaustion. It comes from forcing yourself in one direction so long that you forget all of the other things that are fulfilling to you. It takes time for all of those old desires to percolate up to the surface where you can recognize them now that you're no longer focused on that one thing. A month is great if you're balanced but exhausted and need a vacation since you're not trying to change your thinking habits. It just takes longer to recover from burn out because changing your state of mind is exactly the point.
I just resigned from a 4 year stint in the finance industry--not exactly a startup, but the same elements of meritocracy and self-driven achievement were involved in creating the tunnel vision most here, especially the OP, might appreciate.
The result: I have been left with more "free"/me time over the past 2 months than I can remember. The first 1-2 weeks I slept in till about 11am most everyday, due to sheer exhaustion coupled with the fact that I didn't HAVE to be up and alert for anything/anybody. Since then I have been rekindling old passions and finding a new appreciation for so many things I'd forgotten in the course of my work. I can only hope in the future--when I move on to my next project/job/career---I don't take so many things for granted.
In one month you pretty much have to do the first thing that comes to mind when the month is over.
If you take six months (or better yet a year), you have time to fall in love with an idea, go through the honeymoon stage of prototyping it over a week or two, then discovering you're not into it enough to make it a business. Repeat that cycle a few times, and eventually you'll find something with enough emotional staying power that you can't help but make it your life's work (at least until your next exit).
There's nothing saying this can't happen in a month. But if your attitude is that you're just auditioning ideas during your long time off, you're more likely to make your decision because of fit, not because of time.
By the way, if you have a significant other who doesn't appreciate seeing you idle, this approach is nice because it's indistinguishable from immediately getting back to work.
Six months is just about enough time off that you really start to miss working.
I'd actually recommend setting off with the goal of staying on the road for an entire year. Just travel. Nothing else. Not even a laptop along.
The cool thing about doing it this way is that you won't make it. You'll have the time of your life for 6 months or so, but then a little voice in the back of your head will start asking for more. 8 months in, your brain will just be crying out for stimulation that it can't get from figuring out which local bus to take from here to there. You'll need to work. You'll go out of your way to find a new project, to the point where you'll book a flight home so that you can get moving now.
That's what you need to recenter yourself. If you just take a month-long break, you'll never truly get off your starting point, so you won't end up where you need to be.