I watched your video, and have to confess to a bit of a laugh when you picked up the crank, in what looked like a WTH moment :-)
Some cranks have only a very light cap holding the crank arm laterally, so if the pinch bolts loosen, the crank could fall off before you notice. Square taper and one-key release, or other cranks are a different story. Daniel is right though, that this period of looseness may damage the splines on the crank, requiring replacement.
The 25-30 ft lbs torque you quote is for the external Bottom Bracket bearings, seen in the picture. You probably don't have to touch them. The torque setting you're looking for will be on the crank arm that fell off.
I'm not sure the exact crank you have so it is hard to give specific instructions. Have a look at the Park Repair Help page for various cranksets. I'm guessing yours might be the GXP type. That page gives instructions similar to those in the The Big Blue Book of Cycle Repair, 2nd Edition (Calvin Jones).
If you don't have a torque wrench you can tighten the bolts by having a feel for these things (takes experience and taking into consideration the bolt size, thread size, materials etc). You can also be guided by the specified torque settings by gauging how much more torque should be applied to one bolt relative to another. For the next few rides after re-attaching, take the required tool and check the crank regularly, and tighten it if it comes loose. If it doesn't stay after doing this a few times, you're not tightening it enough, or the crank or BB may be damaged.
Worst case scenario for over-tightening is you will strip the threads on both bolt and crank or BB, and / or shear off or burr the bolt. This will obviously damage those parts requiring replacement and it may be difficult to remove the parts. This could involve drilling out the bolts and various other hacks and so cost a fair bit of labour and parts in a shop - much more than just tightening the crank arm for you which should be very cheap. Thus if you're not confident, $10 or so at the LBS to re-attach your crank is probably money well spent.
Even if you take it to a shop who uses a torque wrench (not all do), and they tighten to the specified setting, your crank can still come loose. I have a great LBS and even using the correct torque settings, I once had the crank arm come loose. The LBS replaced the arm for me under warranty as they followed the manufacturers settings (Shimano). So even if taking it to a shop, I'd still carry a tool and check occasionally, at least for several rides afterward until you feel confident it won't loosen again.
Your crank arm and / or bottom bracket spline could be damaged. You can check for signs of any burring on the splines of the crank and BB, or your LBS can check. Get them to point out any damage to you if they think it needs replacing.