Because he's not terribly self-aware.
Scamander falls into the "absent-minded professor" role. He knows a great deal about magical creatures ("book smarts"), but lacks basic awareness and common sense ("street smarts"). This is a standard character archetype which you will see in many different works of fiction. The story is largely about Scamander growing and becoming a more rounded character. The quest to recapture all of the creatures is a vehicle for this character growth:
- During the bank scene, it becomes painfully obvious that he had no plan whatsoever for any of his creatures escaping. He simply did not expect this to happen at all, and handles the situation very poorly (disillusioning himself, for example, would have gone a long way).
- In the scene with the Erumpent, he displays a greatly improved ability to improvise, but is still largely relying on book rather than street smarts (he's focusing on the Erumpent mating dance, not on the fact that Jacob is about to get impaled and/or blown up). We see the same thing during the department store scene (his primary function is as an encyclopedia, while the other characters do the "work").
- The escape from MACUSA marks a shift: Scamander's situational awareness is greatly improved and he is able to act adroitly and independently.
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By the end of the movie, he is able to identify an undercover GrindelweldGrindelwald largely on intuition and an offhand remark by the latter.
At the beginning of the movie, Scamander simply does not have the common sense to realize that his plan is fragile and unlikely to succeed. His creatures come with him across the Atlantic, not because he planned it that way, but because he failed to make alternative plans for them.
It could be argued that there were no reasonable alternative plans to be made, but I must disagree with that. For example, Scamander could have easily left the case with Dumbledore, along with some simple care instructions (though he would then need some alternative means of transporting Frank to Arizona and/or wherever he was really going). If he did not feel comfortable with Dumbledore caring for them, he could have purchased a more secure carrying case, perhaps one which would only open to his touch, or with some other security mechanism (e.g. "Doesn't pop open by itself"). If even that was too much, he could have acquired a cheap padlock or (maybe) cast Colloportus on the case, or even wrapped a bit of twine around it as he does at the end of the film. He did none of these things.