Until recently, D-Wave's quantum annealing devices always started from a uniform superposition over all $N$ qubits:
$H_{initial} = |+\rangle_0 \otimes |+\rangle_1 ... \otimes |+\rangle_N$
where $|+\rangle_i = \frac{1}{\sqrt{2}} (|0\rangle_i + |1\rangle_i)$.
So let's suppose you already ran a few anneals with this setup and one of the low-energy results looks like a relatively good solution (some local optima) to your optimization problem. Until the very recent introduction of the reverse annealing feature, it was impossible to use this solution as input for the next anneal in order to explore the local space around that solution for bitstrings with even lower energy. Hence, reverse annealing allows us to initialize the quantum annealer with a known (classical) solution and search the state space around this local optima.
When exploring complicated (rugged) energy landscapes of optimization problems you need to balance the global exploration of the state space with the exploitation of local optima. In traditional (D-Wave) quantum annealing, we start with a high transverse field which then gets gradually decreased as you described in your question. D-Wave's quantum annealer was thereby performing a global search (due to a lot of quantum tunneling) at the beginning of the annealing schedule when the transverse field is strong. As the transverse field gets weaker, the search gets more and more local. In contrast, reverse annealing starts with a classical solution defined by the user, then gradually increases the transverse field (backward annealing) to then decrease the transverse field again (forward annealing).
This introduces the new parameter reversal distance which determines how far you want to anneal backward (how strong the transverse field should become). D-Wave published the following two plots in this D-Wave Whitepaper:
In the left plot you can see that reversal distance is a very important new hyperparameter since its value determines the probability of obtaining a new ground state (blue region). If the reversal distance is too low, you will get the same state you started with (red region) which would be useless. And of course if you reverse anneal for too long you essentially perform traditional quantum annealing and loose the information that you started with. Remember that too much transverse field means that we are performing global search again!
The right plot shows essentially the same thing by plotting Hamming distance against reversal distance and the probability of obtaining a new ground state. For your problem at hand, you want to find that sweet spot (maxima of the red curve). For large reversal distances we again see that we get solution strings that are far from our initial state in terms of Hamming distance.
All in all, reverse annealing is pretty new stuff and to the best of my knowledge there are no published papers about its effectiveness. In their Whitepaper, D-Wave claims the generation of 'new global optima up to
150 times faster than forward quantum annealing'.
