It means that with the modification you can connect up a battery (e.g. LiPo) to power it, but you would still need an external charger.
I'm not sure what that modification specifically is, but another option is to use the VIN connection mentioned in the third-from-the-bottom bullet-point on the first page. This VIN method would still work with a single LiPo, but not as optimal. However the VIN method is also useful if you wish to power it from (say) two coin cells or a 9V battery (as examples). I've not tried this so far, but I don't see why it wouldn't work.
In any case, regardless of the method, any charging functionality (if the selected battery is rechargeable) would have to be separate, it is not part of the existing functionality.
In the document is written "tinyTILE may be configured for use with an external battery and battery charging management controller by moving a zero-ohm resistor on the board. This requires unsoldering and soldering the jumper".
This means that is possible to charge an external battery ?
