The SARSA algorithm is a stochastic approximation to the Bellman equations for Markov Decision Processes. One way of writing the Bellman equation for $q_{\pi}(s,a)$ is:
$$q_{\pi}(s,a) = \sum_{s',r}p(s',r|s,a)( r + \gamma \sum_{a'}\pi(a'|s') q_{\pi}(s',a'))$$
Where $p(s',r|s,a)$ is the probability of transition to state $s'$ with reward $r$ given current state $s$ and action $a$.
In Dynamic Programming solutions to MDPs, the Bellman equation is used directly as an update mechanism that converges to correct values for $q_{\pi}(s,a)$. For estimating the value of a fixed policy, this works because the only stationary points of these updates are when the two sides are equal, and there is one equation for each $q_{\pi}(s,a)$ relating it linearly to one or more other $q_{\pi}(s',a')$, so it has a computable solution.
When you add the search for an optimal solution, such as in SARSA, then the policy $\pi$ changes. There is a proof that changing the policy to pick the action $\pi'(s) = argmax_a q_{\pi}(s,a)$ will always either improve the policy or be optimal. This is called the Policy Improvement Theorem, and is based on the inequality $v_{\pi}(s) \le q_{\pi}(s,\pi'(s))$ - there is an extension to the theorem that covers $\epsilon$-greedy policies used in learners such as Monte Carlo Control or SARSA.
TD learning, including SARSA and Q-Learning, uses the ideas of Dynamic Programming in a sample-based environment where the equalities are true in expectation. But essentially you can see how the update $q_{\pi}(s,a) = \sum_{s',r}p(s',r|s,a)( r + \gamma \sum_{a'}\pi(a'|s') q_{\pi}(s',a'))$ has turned into SARSA's update:
The weighted sum over state transition and reward probabilities happens in expectation as you take many samples. So $Q(S,A) = \mathbb{E}[ \text{Sampled}(R) + \gamma \sum_{a'}\pi(a'|S') q_{\pi}(S',a')]$ (technically you have to sample R and S' together)
Likewise the weighting of the current policy happens in expectation. So $Q(S,A) = \mathbb{E}[ \text{Sampled}(R + \gamma Q(S',A'))]$
To change this expectation into an incremental update, allowing for non-stationarity as the policy improves over time, we add a learning rate and move each estimate towards the latest sampled value: $Q(S,A) = Q(S,A) +\alpha[R + \gamma Q(S',A') - Q(S,A)]$
For a more thorough explanation of the building blocks of algorithms like SARSA and Q-Learning, you can read Reinforcement Learning: An Introduction. Or for a more concise and mathematically rigorous approach you can read Algorithms for Reinforcement Learning.
