How to find the minimum value of this expression?

I assume that the $x_i$ and $b_i$ are restricted to positive values? Also I'm assuming that you meant to write $\prod(1/b_i)^{b_i}$ for the guess.

You can find the result you guessed in much the same way that you found it for $n=2$. Setting the derivative with respect to $x_i$ to zero yields

$$x_i=b_i\sum_j x_j\;,$$

which is a homogeneous system of $n$ linear equations for the $n$ unknowns. Due to the condition $\sum b_i=1$, the rank is $n-1$, so there's a one-dimensional subspace of solutions, namely $x_i=\lambda b_i$ with arbitrary $\lambda$. The parameter $\lambda$ drops out of the function to be minimized, so you can take $\lambda=1$, i.e. $x_i=b_i$, which yields the minimum value you guessed.

I assume all $x_i$ and $b_i$ are positive. Put${x_i\over b_i}=:\lambda_i > 0$ and $\prod_i b_i^{b_i}=:B$. Then by the AGM inequality one has $$\prod_i x_i^{b_i}=B\ \prod_i \lambda_i^{b_i} \leq B\ \sum_i b_i\lambda_i = B\ \sum_i x_i$$ with equal sign iff all $\lambda_i$ are equal. This immediately leads to the conjectured result.