The cited sentence
To defray the publication cost, authors are requested to arrange print charges of their accepted papers at the rate of US$ 40 per page from their institutions/research grants, if any.
asks authors of accepted papers to arrange for their institutions (if they are affiliated with one) or their research grant (if they have received one) to pay the journal $40 per print page (meaning, presumably, per page of text appearing in the journal) to defray publication costs. So "if any" means, in the first instance, "if they are affiliated with any institution," and, in the second instance, "if they have received any research grant."
The quoted language does not address situations in which authors are neither affiliated with a particular institution nor the recipient of a research grant. My guess is that either the journal has a separate provision—detailed elsewhere—to cover institutionless, grantless authors or such authors get a free ride. It could also be that the journal almost never accepts papers from institutionless, grantless authors.