The Time Traveller is in his home, speaking to a group of men that includes the narrator. He is lecturing on the fourth dimension. He tells them that a cube exists not only in space, but also in time. Time is the fourth dimension. Many of them are skeptical. The Time Traveller claims that one should be able to move about in the fourth dimension just as one can move about in the other three. After all, he notes, we are constantly moving forward in time, why not move faster or slower or even backward? He produces a miniature time machine, the size of a clock, made of ivory and crystal. The Time Traveller explains that one lever sends the machine into the future and the other one sends it into the past. He asks one of the guests to push the forward lever, and the machine disappears in a small gust. He claims that the machine is now gliding forward into the future. The guests ask why they cannot still see it, since they too are moving into the future, and the Time Traveller explains that it is moving forward too quickly to be seen, like the spokes of a wheel or a speeding bullet. The guests are amazed. The Traveller then shows them a much larger machine, with which he plans to explore time.
The narrator concludes that not many of the guests believed the Time Traveller, as he was a very intelligent man, likely to play elaborate pranks. The narrator returns to dinner at his house the next week. The guests include some of the men from the previous week and some new guests. They have been instructed to begin dinner without their host. When he enters, he is incredibly dusty and dishevelled. He quickly drinks some champagne, then goes to wash up. The narrator suggests to the other guests that their host has been travelling in time. The others are incredulous and make sarcastic remarks in reply. When the Time Traveller is finally ready to tell his story, the guests quickly raise objections. The Time Traveller says that he has no energy to argue and will speak only if everyone agrees not to interrupt. The guests agree, and sit in increasingly rapt attention as the story begins.
In The Time Machine, there is a story within a story. The first two chapters make up the outer story, the frame. What follows is the Time Traveller's story. It is important to consider why Wells included a frame story. It lets the reader know that the story takes place in Victorian England, in a world of gas lamps, cigars, and gentlemen with the leisure time to discuss topics like the fourth dimension. It also sets up a good deal of suspense. The small time machine that disappears could be proof that time travel is possible, but it could also be some kind of parlour trick, an illusion created with mirrors. It opens the reader up to the idea that time travel might be possible. In the second chapter, we see the dishevelled time traveller stumble in. The reader recognizes that he must have been travelling in time. This whets the reader's palate, and also makes the story seem more plausible.
I think it's important to realize that when the time traveller leaves in the end, he has with him a bag and a camera, so the reader can infer that he will return with proof.