Newbie, what do I need besides Espruino WiFi and MPU6050 module?

The fact that it resets the Espruino board makes me think that something might be wrong with the wiring? Bad connections would usually make things like I2C functions fail with an error, but not make the board reboot.

Are there pins that can tighten to fit snug in the holes?

There are, but honestly they're pretty evil. You usually have to clamp them in with a vice, and when I looked at it (I originally wanted to ship Espruino boards like it) I felt it was much more likely folks would damage boards doing that than if they just soldered.

If you're going for it then buy a soldering iron (one with a reasonably thin tip), solder, and then some protoboard and whatever cheap pinned components you can find - you can then do a bit of practice first with stuff you won't care about. After the first few you'll be fine with it :)

Also I'd get a 'solder sucker' - it's dead easy to put too much solder on when you're starting off, and it'll be a pain trying to get it off. For a few dollars a solder sucker will easily take off extra solder if it all blobs together.