By the phrase "fill up with snow," the speaker wishes to convey that it is snowing, and heavily enough to "fill" the woods with snow. By this he means both that the sky is full of snowflakes and that much snow will end up piled on the ground.
The full stanza is as follows:
Whose woods these are I think I know.
His house is in the village though;
He will not see me stopping here
To watch his woods fill up with snow.
The owner of the woods won't see the speaker because this owner is in his house in town, far away.
When the speaker says he will watch the woods fill (the woods will fill slowly) up with snow, this also suggests that he will stop for a relatively long time to watch the snowfall. This is indicated as well by the fact that his horse starts to shake his harness bells later in the poem, wondering why they are not moving onward.
The speaker doesn't often get a chance simply to stop and contemplate the beauty of snow falling against the night sky in a silent woods. He does have to complete his journey, but he nevertheless, for once, takes time out to watch a beautiful natural scene.