Likely, the 12-volt battery was totally dead and the original jumper cable connection was poor. You had too little amperage to start the car. The clicking sound is usually the starter selenoid.
Normal starters, unlike the built-in booster motor in the Insight, are normally disengaged from the engine. Turn the starter switch with the key and an electro-magnet pulls a plunger (the selenoid) in one direction in a tube against a spring that normally holds the plunger in the other direction.
The plunger shoves the starter's gear into mesh with the teeth on the edge of the flywheel of the gas engine. At the same instant, this plunger also throws a Really Big Switch that carries the significant current to run the electric starter motor.
Typically, this "switch" consists of a big, fat washer that gets shoved up against two large bolt heads which act as contacts to carry the electric current to the starter motor.
The electric motor turns, the car starts and you release the starter switch. The plunger's spring retracts the starter's gear, freeing the link between the engine and the electric starter motor and turning "off" the switch sending electricity to the starter.
Too little amperage and the plunger toggles back and forth, making a clicking noise. My guess is that the selenoid has enough "juice" to throw the Big Switch and engage the gear, but the drain of the actual starter motor sucks the voltage so low that the selenoid then lacks enough power to hold the plunger against the spring, so the selenoid spring retracts the plunger, turning off the Big Switch, but then the load of the electric starter motor is gone, so the voltage comes back up and the selenoid throws the plunger again, turning on the starter motor, which drains the voltage again... Tada! Clicking sound. An analog voltage meter hooked up to the 12-volt electric system on the car would look like a mechanical metronome, counting the rhythm of the clicks.
That clicking sound can also happen if the contacts in the selenoid's plunger (the two bolt heads) are burned away from years of use in a conventional engine, but that will likely never happen in a Honda Insight, since the conventional starter is so rarely used.
I know this because the contacts burned out on my 1967 Pontiac Tempest. This taught me a LOT about selenoids. My step-brother "jumped" the selenoid with a screwdriver while I turned the key inside the car to start the car so we could drive it up onto ramps to work on it. We then disassembled the selenoid. That design left some of the contacts outside of the perimeter of the connecting washer, so we could rotate the contacts and put everything back together again and the car started normally for a while, until these relatively small contacts burned away. They lasted a few weeks.
This gave me time to order and get in a new selenoid. Soon, the jury-rig failed and we replaced the selenoid.
