In the context of decision theory (and reinforcement learning, which is the trendier name for this field of research nowadays), the Bellman equations are the most important equations because all algorithms used in reinforcement learning are derived from them. Even though they might not seem groundbreaking they are the foundations on which the whole field rests.
Let's take as an example the policy iteration algorithm. It consists of alternating a policy evaluation and a policy improvement step until convergence.
If we take the Bellman equation for the value function, we have
$$V^\pi(s) = \mathbb{E}_{a_t \sim \pi(s), s_{t+1} \sim P(.|s, a_t)}[R(s, a_t) + \gamma V^\pi(s_{t+1}) | s_t = s]$$
where $\pi$ is the policy, $P$ the transition probabilities and $R$ the reward function.
You can calculate this exactly in the discrete case, if $P$ and $\pi$ are known, because it's a system of $|S|$ (the dimension of the state space) equations with $|S|$ unknowns (the $V^\pi(s)$).
Once you've evaluated your policy, you perform a policy improvement step which consists in setting (pay attention to where $\pi'$ and $\pi$ are used) $$V^{\pi'}(s) = \text{argmax}_a Q_\pi(s, a) = \text{argmax}_a \mathbb{E}_{s' \sim P(.|s,a)}[R(s, a) + \gamma V^\pi(s') | S_t=s, A_t=a]$$ The fact that the new greedy policy $\pi'$ is a strict improvement over $\pi$ can be derived from the Bellman equations (you can refer to the Sutton&Barto book section 4.2 for the full proof). It's called greedy because we've assumed that by changing $\pi$ by only looking at what's better one step ahead we've improved $\pi$ for all following steps.
The final piece of the puzzle is that we've designed a sequence of monotically improving policies. We know it converges towards the optimal policy in a finite number of steps because there's only a finite number of policies in the discrete case.
Hopefully this gives you an idea of why the Bellman equations are so important.
