I am a complete layman in linguistics, so the question I have probably has no scholarly merit whatsoever. My question concerns the definite articles in Indo-European languages. Almost all Indo-european languages spoken in Europe have both definite and indefinite articles - for instance, all Germanic languages and Romance languages have them (correct me if I am wrong). Whereas, none of the languages spoken in India that belong to this family have any articles in the same sense as in, say English. Yes, there are determiners like 'one' in those languages which may be considered a kind of indefinite article, but that is debatable. So, is the emergence of definite articles a phenomenon exclusive to European tongues following their split from the Indian? Or is it that the articles in the Indian languages somehow got lost down the line?
You're forgetting Slavic languages which generally don't have definite or indefinite articles (with some borderline exceptions) and Lithuanian. You can see the spread of definiteness marking in the WALS map.
The etymological origins of definiteness marking are quite easy to identify.
Definite articles generally originate in some sort of demonstrative pronoun. And indefinite articles often come from words like 'one' or 'some'. It's also not difficult to imagine why they might be found useful by speakers. In fact, in modern spoken Czech you can sort of see incipient use of demonstratives to mark definiteness in some situations.
But that doesn't answer the question of why definiteness marking has developed in some languages and not others.
Here the most likely answer is areal diffusion.
While it is likely that definiteness marking in disparate languages has evolved independently, it is also likely that the clusters of use of articles are due to areal influences.
A good example of this is the Balkan language area (the famous Sprachbund) where most of its members (except Greek) have a postponed definite article which includes the Slavic members who could not have inherited definiteness historically and the one Romance language where definiteness looks nothing like in the other languages in the family.
Modern Romance and Germanic languages have indefinite and definite articles, but their ancestors Latin and proto-Germanic didn't have them. It's easy to get a feeling for how the numeral one and demonstratives evolved into them.
Suppose we start with a language in which it is normal not to use articles:
(1) There is tree over there. Tree is tall.
This should be familiar because speakers of languages such as Russian often speak English like this. Sometimes you want to stress it's only one tree, and that the tree in the second sentence is the same as the one in the first sentence:
(2) There is one tree over there. That tree is tall.
Now it can happen that over the centuries, the style (2) of expressing things becomes more and more popular until finally using one or that doesn't necessarily stress anything any more and style (1) falls totally out of use. At this point, the pronunciation of unstressed one and that becomes more and more careless, until it is distinguishable from stressed one and that:
(3) There is a tree over there. The tree is tall.
Note that this evolution explains why often there is no indefinite plural article but a definite plural article. For indefinite plurals there are too many choices (two, three, four, ..., some, many, ....) to start the process of grammaticalisation.
In the Romance languages, the indefinite singular articles are still mostly indistinguishable from the numeral one, and the definite articles, typically spelled le or il (masculine) and la (feminine), developed out of the Latin demonstratives ille/illa (that), which became definite articles already in Vulgar Latin.
I am not sure if the parallel development in the Romance and Germanic languages is due to mutual or one-sided influence, but it seems clear that it was probably not independent.
PS: The boundary between articles and noun declensions is fluid. Articles don't always precede a noun. In languages where they follow a noun, they are often written as suffixes. E.g. this is the case for the definite articles in Danish and in Bulgarian. (You can understand this if you imagine a definite article developing out of "Tree over there is big" rather than "That tree is big".) So the main difference between the articles in many European languages and the noun declensions in Sanskrit is that the articles distinguish definiteness and singular/plural, whereas the noun declensions distinguish only singular/dual/plural.
PPS: There is also a typical reverse evolution that makes articles disappear. They merge into the word they refer to and become prefixes or suffixes which due to lack of stress gradually adjust to their environments and develop irregularities or disappear completely. E.g. the definite articles in Danish and Bulgarian are already written as suffixes.
This is an important source for noun endings that indicate person and number as Latin and Greek, and for declination systems. Proto-Indo-European already had these things, so it's likely that an ancestor of this ancestral language had developed postfix articles.
It's a continuous coming and going of articles and declinations. But the process is usually so slow that you have to go back to a pretty unrecognisable stage of your native language before you can see a really significant change in these aspects of grammar.