Why were Germanic tribes allowed to cross the borders of the Roman Empire?

First of all it was not the Germans. Germany and the Germans did not exist back then. It was the Germanic peoples. Germanic is a linguistic group which includes German, Dutch, Danish, Swedish, Norwegian and English. These Germanic peoples were the ancestors of the peoples who speak these modern languages.

The Franks who lived around the mouth of the river Rhine were allowed to settle in Holland south of the river Rhine and in Belgium because they were allies of the Romans and supplied troops and cavalry to the Romans.

The Visigoths asked the Romans to be allowed to settle in the lower Danube area of the Roman Empire because they wanted to escape the Huns who were conquering Eastern Europe. Emperor Valens granted this in 376.

The Ostrogoths settled in Pannonia (eastern Austria and western Hungary) while the Romans lost control of this area to the Huns. After the collapse of the Hun Empire the Romans and the Ostrogoths became allies.

Other smaller Germanic peoples were allowed to settle in smaller areas of the Roman Empire because they needed land and because they could become suppliers of soldiers for the Romans.