Intuition: this problem illustrates a fairly common strategy for constructing a martingale. First, a collection of random variables $(B_n)_{n\geq 1}$ is generated sequentially, with $B_n$ defined in terms of $B_{n-1}$. The method of construction ensures that, though they are not independent, the dependence structure is relatively simple in that they form a Markov chain. However, typically the conditional expectation requirement for a martingale is not satisfied yet, so we seek functions $f_n$ such that the random variables $M_n=f_n(B_n)$ do satisfy the martingale condition. In this case $f_n(x)=\frac{x+1}{n+2}$.
Showing that we have a Markov chain: To show that $(B_n)_{n\geq 1}$ is a markov chain, we want to show that $$P(B_n = k_n | B_1 = k_1, ..., B_{n-1} = k_{n-1}) = P(B_n = k_n | B_{n-1} = k_{n-1})$$ where the numbers $k_i$ are non-decreasing with $k_i\leq i$.
If we condition on $B_1 = k_1, ..., B_{n-1} = k_{n-1}$ then we know that there are $k_{n-1}+1$ black balls in the urn (out of a total of $n+1$ balls) just after time $n-1$. So the conditional probability that $B_n=k_n$ is either $1-\frac{k_{n-1}+1}{n+1}$ (if $k_n=k_{n-1}$ i.e. white ball chosen) or $\frac{k_{n-1}+1}{n+1}$ (if $k_n=k_{n-1}+1$ i.e. black ball chosen). Now, you might ask, what about our knowledge of $B_1$,$B_2$,...,$B_{n-2}$? Well, yes we know them, but their values don't affect the conditional probabilties we just computed. This is just a consequence of the way the sequence $(B_n)_{n\geq 1}$ is constructed.
If we condition only on $B_{n-1} = k_{n-1}$ then... we obtain exactly the same probabilities! The probabilities are the same because $k_1,...,k_{n-2}$ are irrelevant once we know $k_{n-1}$. So $(B_n)_{n\geq 1}$ is indeed a Markov chain.
Further information: can be found in many places online, for example this document gives a good analysis of Polya's urn.
