City is a powerful yet frustrating card. With no piles empty, it's a 5-Cost
Village, but with at least one empty, it's strictly superior to a
Laboratory. So if you're pursuing a City-heavy strategy, you really want to empty a pile quickly. The trouble is that grabbing that last City with your turn's buy means that the first person to get the benefit of the souped-up Cities is your left-hand opponent.
University provides a good way out of this dilemma. If you have a University and a City in your hand when there's just one City left in the supply, you can gain the last City before playing your City from hand, and gain a potentially vital head-start in the monster turn race. This benefit is even greater if there's another useful 5-cost action that your Universities can gain: either Wharf/Council Room/Torturer/Rabble to run a massive +Action/+Card chain, or Bazaar/Treasury/Minion/Market to ramp up your buying power. Once that second pile is empty, your Cities are huge Village-Laboratory-Market behemoths that may be able to give you enough power to end the game on their own.
Example game with MinionExample game with WharfExample game with Bazaar