Why using short traces with the MCP73831 LiPo charger?
Think of the short wire recommendation as a 'best practice'. This application is fairly low current, so the risks in disobeying are somewhat minimized. That being said, here are a few 'best practice' reasons to keep the wires short:

This particular IC doesn't have separate load voltage sensing inputs, so any loss in the cabling will manifest itself as a load setpoint error which varies as a function of the current (higher current means higher \$I^2R\$ losses and lower voltage at the load). 

Another effect: very long cables could have enough inductance to resonate with the \$4.7 \mu F\$ capacitor at pin 3 of the IC, especially if a charged battery is plugged in and the IC has no supply voltage. The IC may get damaged if the voltage on the charging pin gets excessively high.