How does foreign investment solve India's economic problems?

I'll try to answer this as best as I can.

Well, if we are self-sufficient, we need not depend on anyone else for our own progress. That is true for any field, be it education or self-improvement or agriculture, or in this case, national economy.

And sadly, we certainly are not self-sufficient. We need energy, lots of it. And we need a lot of oil too. We can't produce enough oil to satisfy our demand, so we 'buy' them from whoever is selling it. And they don't want rupees in exchange, they want dollars. Or Euros. Or whatever other currency they are trading in. So for paying them, we need to have reserves of foreign currency. This is called <a href="https://en.wikipedia.org/wiki/Foreign-exchange_reserves" target="_blank" rel="nofollow noopener noreferrer">Foreign Exchange Reserves</a> of the nation. This is very important, because without significant Forex reserves, a country cannot participate in international trade effectively. Too much of it is also bad, and that's another thing. But the point is, every country needs to have Forex reserves, and foreign investments is one way you can get foreign currency to your coffers. There are other ways like directly buying US treasury issued bonds et all.. But generally, I think this is why foreign investment is considered good for the economy.

By the way, some interesting links about this whole Forex thing,,

<a href="https://en.wikipedia.org/wiki/Reserve_currency" target="_blank" rel="nofollow noopener noreferrer">Reserve Currency</a>
<a href="https://en.wikipedia.org/wiki/Triffin_dilemma" target="_blank" rel="nofollow noopener noreferrer">Triffin Dilemma</a>

Of course, this is only based on my understanding of economics, which is quite low, as it is. There might be other reasons..