Lets take an example to understand,
You have two banks one is Bank A and another is Bank B. You invested $1000 in bank A and $1000 in bank B where both the banks at 5% interest. Now this is equilibrium.
Assume now bank B increase the interest rate of deposit to 8% from 5% so you as a depositor will withdraw money from A and put it in the bank B.
Now take the case of countries, most of the US investor investment is in the private sector (read as equity or PE) which is expected to give more return then US interest rate. Suddenly US increases the rate of interest and there is no change in the return expectation in private placement in India so slowly the new investment from FII/FDI will come down and existing investor also will pull if it is less attractive then US. So in-short dollar outflow will be there and rupee will depreciate.