"The more than" is a noun phrase whereas "more than" is a comparative adjective.
In the first case, you want to introduce a noun phrase. For instance, you could replace, in the original sentence, "the more than 300 animals" by X(noun phrase) and you will complete the sentence in a grammatically correct way.
eg Releasing X [is good for the environment]
Using "more than", by omitting the definite article "the", you will use a comparative article form which will change the structure of the sentence: You will be comparing X to Y.
eg Releasing more than
300 animals [is more effective than releasing Y] --> more than
here plays the role of an adjective, a comparative one. You then see why this is called the comparative form.