Question: In the following sentence:

> 
> By the year 2050, we will have 10 billion people on our planet. Linking this rise to the ‘what’ and ‘how’ of feeding the entire population in a sustainable manner is what **got Varun Deshpande and Ramya Ramamurthy of the US-based Good Food Institute’s (GFI) India chapter started on creating their new podcast, Feeding 10 Billion**. “Our food system is broken. The greenhouse gases emitted by industrial animal agriculture are more than the emissions by all forms of transportation combined. However, demand for meat continues to rise, and telling people what they should eat or guilting them into giving up meat doesn’t work,” says Ramamurthy, adding that the way forward is to offer people protein made in a better way.
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**got somebody on doing something** I think this usage is wrong, because it is usually used as **get somebody to do something/get somebody doing something**.

Also I don’t understand meaning of the term **India chapter started**

Answer: 
You've mis-parsed this. "started" is part of the phrase you're confused about: it's "got somebody started on doing something". It's not part of "India chapter".

You might rewrite this as:

> 
> Varun Deshpande and Ramya Ramamurthy are members of the India chapter of the US-based Good Food Institute’s (GFI).
> 
> 
> Linking this rise to the ‘what’ and ‘how’ of feeding the entire population in a sustainable manner is what got them started on creating their new podcast, Feeding 10 Billion.
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[...] got Varun Deshpande and Ramya Ramamurthy of the US-based Good Food Institute’s (GFI) India chapter started on creating their new podcast, Feeding 10 Billion.

I am removing where these guys work to show the usage:

[...] **got** Varun Deshpande and Ramya Ramamurthy started on[...] creating their new podcast, Feeding 10 Billion.

**got them started on their new podcast** means: caused them to start doing it.

Another example:
Person 1: What got you started on writing?
Person 2: Answer: Reading great authors.

Now: US-based Good Food Institute’s (GFI) India chapter

means: The India Chapter of the Good Food Institute.

written in the question in the possessive form.

The phrase is: to get started on doing something

I got started on answering this question five minutes ago.