Well, fish produce ammonia like we produce urine. They release that ammonia into the water through their gills. Ammonia is toxic to the fish (hence the fact that they get rid of it through their gills). So, we need a way to get rid of that ammonia.
Fortunately, nature has taken care of that for us. There are certain types of bacteria that consume ammonia. Those bacteria are all around us, and over time they will grow in large numbers in your filter and consume all the ammonia your fish produce.
This is all well and good, but when they consume that ammonia, they release nitrite, which is also very toxic.... But again, nature has it covered, and there is another type of bacteria that consumes nitrite. Those bacteria then release nitrate. Nitrate isn't great, but it's a lot better than either ammonia or nitrite.
These bacteria don't just appear overnight in your filter though. There are a few there to start with since they're all around us. However, to start with, there aren't anywhere near enough bacteria to consume all the ammonia your fish will produce. It takes time for the bacteria to reproduce enough to have a colony big enough to "eat" all the ammonia. Once the ammonia eating bacteria have grown big enough, the nitrite bacteria have to run. This whole process can take a few weeks.
The good news is, if you're keeping the same filter that has been running on the tank, then you already have those bacteria colonies. However, some die-off will occur during the transport process, so there will be a small spike of ammonia and nitrite as the bacteria "re-grow".
Since you will already be seeing a small ammonia spike, you don't want to make it worse by adding more fish which will just increase the amount of ammonia being added to the water.