What does the part in boldface mean? "saw mountains a short distance across the sea."
Those words are being used in a fairly standard way. It means: they saw mountains. They saw mountains, across the sea. They saw mountains, on the other side of the sea. They saw mountains, a short distance away, on the other side of the sea.
Does it convey there were two ranges of mountains on the two sides of the sea?
No, the sentence only mentions one range of mountains, not two. (However, the entire source text which you haven't included here could discuss a second range of mountains...)
how does what the sailors saw proves the fact stated in the next sentence.
The very brief snippet of information doesn't prove the fact. You could see mountains across a sea, and discover that at yet a further distance, a hundred miles away, the land masses are attached and are part of the same continent. However... if Bering sailed extensively around Russia from the Arctic Ocean to the Pacific Ocean then the facts would be apparent.