I'm not sure how to get from the left hand side to the right. It appears in Sipser's Introduction to the Theory of Computing, in a proof of showing that a $t(n)$ time nondeterministic single-tape Turing machine has an equivalent $2^{O(t(n))}$ time deterministic single-tape Turing machine. 
Why is $O(t(n)b^{t(n)}) = 2^{O(t(n))}$?
complexity theory;time complexity