What does the traveller think is the only way to discover the time machine? What was the result of his restlessness
By discover, I think you mean whether the time machine actually worked. He built a device that he says will travel through time, and his friends, his dinner companions really don't believe him. Even after he shows them the demonstration with the miniature time machine. In order to prove that the machine actually works, the time traveller decides to use the machine and travel forward in time.
His restlessness has two sources, one, he wants to try his time machine, and two, he longs to escape from his own time because he believes that it is too consumed with war and violence and inequality. So his restlessness leads him to decide to take a trip into the future in search of a time where there is no war or poverty, or hunger, he is looking for a utopia.
"the Time Traveller's experiences showed a future of doom, as his journey revealed a world in which the struggles of the 1890s were not resolved but rather exacerbated. His journeys even deeper into the future revealed a world in which humanity had been extinguished from the face of the earth."
He travels forward far into the future to 802,701, and discovers what he believes is paradise, but there is a dark side to the developments of humanity over the thousands of years he has traveled.
After he discovers how the society actually operates, he is desperate to escape.