There are many ways they can disrupt their new environments.
They can often carry diseases or parasites that native species are unequipped for. A good example of this is cats being brought to Australia, a continent with no native felidae. Cats are one of the main carriers of toxoplasmosis and many species in Australia had not evolved alongside felidae and were relatively unequipped to handle these parasites if their immune system was vulnerable.
They can also cause issue simply because they are so different, and the animals and plants living I. These areas are simply unable to defend themselves or keep up as a result. Such as pigs (and other animals brought over by humans) eating dodo bird eggs (these birds roosted on the ground) and domestic cats overhunting for sport.
Hope this helps a bit! It's a pretty broad topic and definitely one worth researching even more