They lie because they need votes, it's as simple as that.

Let's take a country in which the whole system got corrupted. Imagine you are a politician and were elected by people after giving certain promises.
Let's say you promised people that you would build a hospital having 2000 patients as capacity.
The money required for building such a hospital would be 20K$.You've now requested the government to release the fund.
Head officials would take 15% and give pass the rest.
District or province officials would take 5%.
You, if honest would not take any but if not, 5% drops into your pocket.
The contractor who builds the hospital would take 5%.
The rest amount would be 14k$ and with that you might not be able to build the desirable hospital. You haven't stood on your word, it might be because of you or the other person but the promise is broken.
Some of the plans would not even execute because of corruption.