The Huns drove them out of their lands. The Goths needed to find better lands to support their population. The Roman Empire was weak. The Romans had better land. The Romans couldn't defend their land against the Goths and the Goths couldn't defend their lands against the Huns.

As for the climate change argument this movement of people groups is not uncommon. Rome up until the arrival of the Goths had been dealing with "barbarian" incursions for a thousand years.

Rome invade Gaul, Germany, and Britannia which people seem to omit. There is no argument that climate change forced them too. They did it because they could. The "barbarian" tribes invaded because they could. Both sides in this conflict are human and have human motivations.