It means 'reveal something that is hidden or secret'
The river freezes, and snow falls on the ground like a blanket covering the countryside. However, a river surface is flat, while on the land there are bushes, rocks, ditches, and small rises (like bumps on the surface of the land). So, curving through this rumpled blanket of snow on the land is this perfectly flat, smooth surface of the river ice on which the snow rests, also perfectly flat. It is this flatness, this smoothness of the snow which gives it (the river) away - betrays/reveals that there is a frozen-over river under the flat snow even though it is hidden by the snow.
I read this from The Tenderness of Wolves by Stef Penny:
"The smoothness of snow on river ice gives it away; a wide road curving through the plain six or seven feet below ground level."
What does "give it away" mean here? Give what away?
The smooth snow indicates that it landed on a smooth surface (the frozen ice of the river). The surface of the land around the river ice would be rough and uneven as it covered the plants and rocks. So, 'give it away' in this instance means that it is plain to see where the river ice is located.