How can the SparkFun Lipo USB charger power other devices at the same time as charging?
[first things first: read the MCP73831 datasheet, of you haven't done so already]

This is a fairly simple charge controller.  The load (SYS_OUT, aka SYSTEM_LOAD) is wired directly to the battery, so it draws current from both the battery and charger.


"Couldn't this ruin things?"  That alone shouldn't ruin things.  The charger is current-limited.  Of course, it also depends on the specific nature of your things, which we don't know enough about.
Max voltage that MCP73831 outputs is 4.20V.  The load has to be designed to accept such voltage (that's not difficult to achieve).
If the battery is completely discharged, then the load will not power-up until the battery gets some charge.
The battery will be charging more slowly when the load is consuming part of the charge current.
Why hard-wire the load to the battery?  Because that's simpler.
More sophisticated charge controllers have a feature called PowerPath, where the load is not hard-wired directly to the battery.  (STNS01 is an example. See block diagram on p.4 of the STNS01 datasheet.)