Yes, Crossbow Expert lets you attack with a crossbow more than once a turn.
There are two weapon properties at play here - Ammunition and Loading.
Ammunition. You can use a weapon that has the ammunition property to make a ranged attack only if you have ammunition to fire from the weapon. Each time you attack with the weapon, you expend one piece of ammunition. Drawing the ammunition from a quiver, case, or other container is part of the attack (you need a free hand to load a one-handed weapon). At the end of the battle, you can recover half your expended ammunition by taking a minute to search the battlefield.
This factors into the crossbow conversation mostly due to the sometimes confusing bit about one handed weapons. The confusion comes from trying to apply it to two-handed weapons; it is quite specific about one-handed weapons.
The idea behind it is that you need one hand for the weapon and one for the ammunition, preventing you from holding a shield or second weapon. For two-handed weapons, the second hand involved in firing the weapon can also be used in the loading process.
Loading. Because of the time required to load this weapon, you can fire only one piece of ammunition from it when you use an action, bonus action, or reaction to fire it, regardless of the number of attacks you can normally make.
This property (not Ammunition) restricts the rate of fire. This is also the property that Crossbow Expert removes:
You ignore the loading property of crossbows with which you are proficient.
Additional confusion enters when people try to use two hand crossbows to interact with the third part of Crossbow Expert:
When you use the Attack action and attack with a one-handed weapon, you can use a bonus action to attack with a hand crossbow you are holding.
The second crossbow takes up a hand and prevents reloading (per Ammunition). However, a hand crossbow itself is a one-handed weapon, which is an acceptable weapon to trigger that third clause and allow the bonus action attack. In other words, a character with Crossbow Expert and one hand crossbow can always fire twice a round (once with the Attack action, once with the bonus action), reloading between each shot (per Ammunition).
Characters with a hand crossbow and the Extra Attack feature can get even more shots with the Attack action, but only one from the third function's bonus action. Light Crossbows and Heavy Crossbows do not trigger the third function of Crossbow Expert (because they're two-handed weapons), and only get extra attacks from the Extra Attack feature.
I didn't touch on Crossbow Expert's second function, no-disadvantage-in-close, because it is not relevant to rate of fire.