I think this sentence is understandable, but it has a little difference with:
Let me explain why. First start with the simple one.
我快到中国了 ----> I will arrive in China soon.
This means you are out of China, and will arrive soon. In this time, you are in a peace mode. But If you use
我快到中国啦!, this may indicate you are excited.
我终于到中国了 ----> Finally, I arrived in China.
This means you are now in China. Also, this sentence means maybe you are blocked when you are trying to enter China, or some other reason, but you finally made it. Some similar sentence like "I finally worked the issue out. 我终于把这个问题解决了"
Second, combine the two together.
我终于快到中国了 ----> Finally, I will arrive in China soon. (Maybe my translation is not right.)
This also means you are still not in China, but you have some trouble while trying to enter China. For example, a Chinese is in US, he is trying to fly to China. But some policy don't allow him to fly to China directly. He has to fly to Australia first, and then fly to Thailand, and then to China. So when he is arriving in China, he can say "我终于快到中国了", with tears running.