Apply multivariable linear regression to a dataset in pandas with sklearn
The form of your model is incorrectly specified. You want to model your data so that your independent variables are year (a numeric variable) and state/union (a categorical variable). Your dependent variable will be the population size. 

In order to develop such a model you will need to restructure your data so that it is in a long format (see below)

States/Union Territories      Year                Pop 
----------------------------------------------------- 
Andaman and Nicobar Islands   1951                 31
Andaman and Nicobar Islands   1961                 64
           ...                 ...                ...
Andhra Pradesh                1951              31115   
Andhra Pradesh                1961              35983
           ...                 ...                ...   
Arunachal Pradesh             1951                307
           ...                 ...                ...


Once the data is transformed into a proper format, you can then fit your model.
I should mention you will need to recode your categorical variable - you can use a method called one hot-encoding to achieve this. 

Once your model has been fit, you will be able to predict the expected population size for a given year and Union/State.

The following post Multiple Linear Regression Model in 7 Steps with Python provides a great guide to developing a model.

Hope this helps!