What does it mean to "kickstart the economy"? This answer claims that:


  the central banks have been lowering and lowering [interest] rates, desperate to kickstart the economy, but it's not been working


What does "kickstart the economy" mean here? What exactly are central banks desperate to achieve?
In short, the belief this reflects is that cheap money is spent more freely.  Since GDP counts the sum-total of value-added transactions in the economy, in principle this means that monetary policy affecting the velocity of money can in turn affect GDP.  Theoretically, this is valid to the extent that debt-servicing costs are a significant friction in the economy.

Thinking of money as blood in the circulatory system of the economy, lowering rates is like taking blood thinners to avoid clots.

Whether it works out that way in practice or not is difficult to ascertain, and in practice there are countervailing forces (lower interest rates discourage household savings, high inequality suggests a high degree of misallocation, etc.) that simple changes to interest rates cannot address.