How to interpret ANOVA results?
f_val is the F Statistic value. Mathematically it is

$ F = \frac{MS_{Between}}{MS_{Within}}$

The null hypothesis for your ANOVA is

$H_0: \mu_{Entire home/apt} = \mu_{Private room} = \mu_{Shared room} $ which means all means ($\mu_i$ s) are equal and there is no need of grouping using explanatory variable

vs 

$H_A:$ At least one $\mu_i$ is different. There is a need for grouping

The p-value for this test was very very low, hence python returned 0. Anything less than 0.05 is considered low enough to reject the null hypothesis.

Independent variables are also called explanatory variables. I believe price is a dependent variable.