> and I have no idea where to begin explaining that it's nothing like that
(I have studied Sanskrit and Hindi in school for multiple years, and can understand a good amount of Marathi and Tamil.)
You can say this - I think it is true, not quite verified (partly heard from older relatives, partly deduction based on observing similarities in the mentioned languages' vocabularies):
Sanskrit came first - thousands of years ago. It is the language of the Vedas, 4 of India's most ancient scriptures. Many northern (including east and west) Indian languages are similar to Sanskrit and likely are descendants of it. Hindi, Gujarati, Marwari, Punjabi, Bengali, Oriya (not sure, but it sounds some like Bengali), Marathi, are some of these.
Languages of Indian states further east than Bengal, called the North Eastern states, like Assam, Nagaland, Manipur, etc. may not show a lot of Sanskrit influence because they may have Oriental influence instead, such as from SE Asia, China, etc.
Except Assamese may have some Sanskrit influence, also not sure of Manipuri.
And of the southern states in the peninsula, Tamil has the least Sanskrit influence and is also said to be very ancient. The other three, Malayalam, Kannada and Telugu, have some Sanskrit influence.
Caveat: I am not a language expert.