The eyepiece is used as a magnifying lens. That is, it is used to look at the image from the eyepiece lens. Thus the image produced by the objective lens becomes the object of the eyepiece lens.
For the eyepiece lens, it is placed such that the distance from the image produced by objective lens to the eyepiece lens is shorter than the focal length of the eyepiece lens. Thus what one sees when looking through the eyepiece lens is a virtual image (a magnified image) of the distance object.
It was easy to produce even larger images by simply replacing the eyepiece lens with another eyepeice lens that was more curved (a stronger magnifying glass).
The microscope uses the same idea, except now we wish to observe objects close up. So the objective lens is a small lens which is highly curved (so it can bend all the light coming from the close object). Again, the eyepiece lens is just magnifying the image produced by the objective lens.