how does organic farming does away with pesticides?

There are several practices that organic farmers employ to help control insects and weeds. to name a few, crop rotation, catch crops, hand weeding, hand picking off insects, use of mulches, and many others including just putting up with a few more weeds and insects. But to answer your question as how an organic farmer does away with pesticides, he just doesn't use them.

In general the aim of organic farming/ gardening is crop management without the chemicals. It relies on continuous monitoring of a crop and addressing issues before problems arise in a natural way. Some tools are the use of naturally occuring chemicals that plants produce to keep insects from damaging them, or using those plants themselves as a protection. Also employed are tactics like companion planting and rotation of crops to minimise pest and disease populations, and insects for their control by predation on those who damage our crops. Sometimes microorganisms are used to combat other micro's as well as insects. Simple non toxic chemicals like insecticidal soaps, simple salts, and natural minerals are also used, their timing adjusted because of close monitoring of crops. Fertilizers are selected that are of natural origins in nature and usually applied in their raw and unadulterated forms. In this way we mimic nature. It should be noted though that not all organic methods are in keeping with the theme of sustainability and sustainability does not always mean organic. The materials labeled or "licensed" organic (certification is totally a non-government quid pro quo, the cost of which goes to the consumer and another reason one expects to pay more) can and frequently are just as toxic as the chemicals they replace. One major complaint about many of the organic products used in farming is that bringing them together and making them available requires a huge network of processing and shipping at an increasing cost mostly absorbed by the consumer. Another reason one expects to pay more for organicly produced products and this is the energy for the low input farming movement.