The nitrogen cycle takes the ammonia in the water and turns it into nitrate. To do this it needs oxygen and carbonate hardness (KH). In addition to being needed for the nitrogen cycle to function, KH also protects your aquarium from pH swings by absorbing acid. This is called buffering. Once the KH in the water runs out, 2 things happen. The nitrogen cycle stops working and you start to see ammonia in the water, and there is no buffering so your pH can start to fluctuate. One of the reasons you do water changes is to replenish KH so it doesnt run out.
The natural processes in an aquarium tend to acidify the water. Nitrate in the water forms nitric acid. Fish respiration puts carbon dioxide into the water which when disolved forms carbonic acid. If their is sufficient buffering this acid gets absorbed and you see little sign of these acids. Once the buffering is gone, your water can become acidic quite quickly.
The flip side of this is that ammonia in acidic water is mostly ammonium and non toxic. So despite the nitrogen cycle ceasing to function and you starting to see ammonia build up the fish will be fine. This is commonly called old tank syndrome, where people generally get lazy with tank maintenance after a while, dont keep up with water changes, and there is a degrading of water quality. But you see no signs of this due to the nature of ammonia toxicity.
What you have to be careful of is then jumping right in and getting on top of your water maintenance. You do a big water change, ammonia goes down, but the water change also replenishes the KH. With buffering now in the water the acidity gets absorbed, pH rises. The ammonia left over becomes toxic free ammonia and you see effects in the fish. Smaller, more frequent water changes are better than bigger ones so you dont see this sudden leap in ammonia toxicity. Products that deal with ammonia toxicity can also help when you start to get on top of water maintenance too.