How to keep vegetables and herbs fresh for longer?
Plants continue living after they have been picked. They respirate, but without the moisture and nutrients they used to get from the soil, their metabolism goes awry and they wilt. 

The only way to stop this process is to blanch and then freeze them. This works well, but needs lots of freezer space. Also, they aren't exactly like fresh, with their texture changing a lot (they get soft, as the ice crystals produced by freezing rip their cell walls to pieces). A few of them change their flavor too. 

If you don't want to freeze, you can prolong their life a bit by slowing their metabolism. 


Keep them at 10 to 15°C below the temperature in which they live/ripen. The lower the temperature, the slower a plant's metabolism is. But you don't want to go too low, as it destroys some flavor components and in general triggers unwanted change in your food. In the fridge, tomatoes get woody and lose aroma, bananas go brown, mangoes lose flavor, and so on. You want to keep these at above 10°C. Fruit and vegetables like apples and potatoes which don't require too much warmth for growth can be kept at lower temperatures, including fridge, without taste loss. 
Keep them away from light. Light is an important environmental factor for plants, and promotes growth. You don't want anything growth-related to happen in your picked plants, it hastens their wilting. 
Give them some humidity, but not too much. You want just enough to keep their slow metabolism going. Give them too much, and you will get faster metabolism, which will make them look better at the beginning (like the salad constantly sprayed in the market), but exhaust the available nutrients sooner, leading to quicker wilting. You also risk mold in high humidity. On the other hand, if they are completely dry, the wilting will occur not because of missing nutrients, but because of missing water. I can't give you a rule of thumb for the water requirements of each plant, as they are very different. They are somewhat related to the plant part: fruits (in the botanical sense, so including eggplants, tomatoes, etc.) don't need much, they are mostly self-contained. The tighter their outer membrane, the less they need; a banana will practically never wilt for lack of humidity. Leaves need more humidity, for example salad or herbs. Other parts fall in between. 
Limit oxygen exposure. As I said above, they respire. Unlike the photosynthesis process which some people confuse with plant respiration, respiration uses up oxygen and produce CO2. Less oxygen means less respiration, so they keep longer. Still, storing them in an airtight humid space is a recipe for mold and rot, so you have to be careful about just how much you can limit the oxygen. 
Buy them as whole as possible. The more damaged the plant, the sooner it will wilt. Pre-cut salad goes bad quickly. Hulled corn doesn't stay fresh as long as corn in the husks. Herbs and salads sold with their roots packed in earth (in pots or just plastic bags) hold much longer than the cut leaves. 
Don't wash them. Many plant parts, especially fruits, have a protective coating. Some of them get wax applied by the producer. Washing this coating off speeds up moisture loss and wilting. Store all your fruit, vegetables and herbs unwashed. Clean them immediately before preparation. 
Isolate them from each other. Some fruit produce ethylene, which promotes growth in other fruit. This brings a special problem: the fruit which produces ethylene is better off when not kept airtight, because the overripening effect of ethylene will be stronger than the preserving effect of low oxygen. 
Carefully remove any rotten exemplars. "One bad apple spoils the barrel" is not just a metapher; very ripe (already going soft) fruit produces more ethylene (see above), and bacterial and mold colonies have it easier to jump to the neighbouring fruit if they start from a "strong base" (a fruit they have already taken over). 
Learn which fruit can ripen off-plant, and buy this type slightly unripe, letting it ripen at your home. But be careful to not get other types of fruit unripe, because it will never taste good. (Applies to botanical fruits, e.g. tomatoes taste better when vine-ripened). 


The perfect solution is a special fridge or cellar with the right conditions (which happen to also be good for cheese and wine). Most of us can't afford that, so you may have to create an environment which creates everything but the correct temperature range for warmth-loving fruit (you can keep apples, etc. in the normal fridge). Some people feel that the better compromise is to keep them at room temperature and have them wilt quicker but preserve flavor, others keep tomatoes etc. in the fridge wher they stay fresh but lose aroma. It is up to your personal preference.