The main purpose of the fabric is to keep the stone separate from the soil. Over time, without a separation layer, the stone can sink into the ground and the foundation soften (particularly on clay ground).
Weeds and vegetation will have a hard time growing beneath the shed due to a lack of light, but as the stone can extend about six inches beyond shed wall, the fabric may help a bit in keeping the weeds down. If used, the landscaping fabric should be a heavy duty material which will allow water to drain away.
Placing the crushed stone layer. Before compacting the stone, rake it as level as possible. To compact the stone you could hire a vibrating plate, but if you need a bit of exercise use a hand tamper to compact the ground. Hand tampers comprise a 10kg weight with an area of approx 4''x4'' (100x100mm) which you raise and drop repeatedly to compact the stone. Pound the stone down until it's compacted and even, remembering to keep checking the levels in the same way that you did for the excavation. Add or remove material where necessary to achieve a level platform.