The meaning of kindly is hopefully clear: "kindly help me" means "please be kind and help me" (or "please help me out of kindness"), etc. It's a word used for polite requests; a bare "help me" is impolite relative to "kindly help me".

If you're asking why kindly is more common in Indian English than elsewhere, it's just one of the hundreds of things that have remained in Indian English long after they have gone out of fashion elsewhere.

Searching Google Books for "kindly do this" and looking through the first ten pages, most results, besides a few from India, seem to be from British and some American books, pre-1920. Examples:

"If our friends will kindly do this for us, we shall feel indebted to them." [The Penny Protestant operative, 1842]

"Would you kindly do this?" [Letter from Florence Nightingale, ≈1886]

"If you will kindly do this, I will be very thankful" [Southern and Southwestern Railway Club, Atlanta, 1914]

"If you will kindly do this I will pay you for the two together" [Anthony Trollope, 1864]

"Would you kindly do this library another favor and again place it under obligation?" [Washington State Traveling Library, 1913]

and so on and on (there are hundreds of results), and most interestingly, one 1886 book showing it must have been standard in England:

The first thing that strikes you on landing in America is the want of deference and courtesy among all classes. Not only from the inferior to the superior, but vice versa also. The maxim noblesse oblige has no sway there. In England, speaking to an equal or a social inferior, "Kindly do this," or "Please give me that," is general. In America the "kindly" and "please" are carefully omitted…

[From context, he doesn't mean it's used only with inferiors, but even with inferiors.]

Anyway, given that kindly was standard, this word for politeness entered India when English did—during colonial rule—and it has stayed on. Why something has continued to exist is not a question that can be answered (inertia?); perhaps the right question is why it went out fashion in the UK and US. (And I'd be interested to learn.) My guess is that either the phrase became clichéd, or such politeness came to be deemed excessive. In the US it seems to have taken on a slightly sarcastic meaning: Wiktionary says

kindly

2. (US) Please; used to make a polite request.

Kindly refrain from walking on the grass.

Kindly move your car out of the front yard. Usage notes

(please): Kindly is used in a slightly more peremptory way than please. It is generally used to introduce a request with which the person addressed is expected to comply, and takes the edge off what would otherwise be a command.

Well, in Indian English it happens to have retained its original meaning, is not peremptory, and is a request rather than an expectation. (And in general it seems safe to assume that Indian English expressions are not sarcastic, and to take them at face value.)