Why would I turn canning jars upside down for 15 minutes after removing from the water bath?
That passage is suggesting an entire canning/processing method, one that may not be safe. It will generally work to create a seal, as they say, but it may not fully sterilize the contents and the seal will not be as likely to hold. I would not follow their instructions, and instead process your chutney according to a trusted canning recipe you find elsewhere.
See for example this USDA source, which says:

Some other methods of sealing jars call for inverting a closed, filled jar of hot product for anywhere from thirty seconds to one hour. (Inverting is turning the filled jar upside down on its lid.) While this inversion process can be successful in producing a sealed jar, it works best with very hot product. Individual variation in practicing this procedure or unexpected interruptions can result in delays between filling jars, getting lids screwed on, and inverting the jars. If the product cools down too much, the temperature of the product can become low enough to no longer be effective in sealing jars or preventing spoilage.
When the inversion process does work, the vacuum seals of filled jars still tend to be
weaker than those produced by a short boiling water canning process. A larger amount of
retained oxygen in the headspace may allow some mold growth if airborne molds contaminated
the surface of the product as the jar was filled and closed. More complete removal of oxygen
from the headspace also offers some longer protection from undesirable color and flavor changes with some types of fruit products. A weak seal may be more likely to fail during storage.