Consider a variable $x_i$. If it appears once, or if it appears twice but in the same polarity (i.e., both times as a positive literal or both times as a negative literal), then we can set it accordingly and satisfy all the clauses that contain it (more formally, any satisfying assignment can be converted into a satisfying assignment in which $x_i$ has this value).
The remaining case is the in which $x_i$ appears once positively and once negatively, that is, there are clauses $x_i \lor C$ and $\lnot x_i \lor D$. These two clauses are logically equivalent to the single clause $C \lor D$, which no longer contains $x_i$.
In this way, we keep eliminating the variables one by one, until we either eliminate all of them (in which case the formula is satisfiable), or we reach an empty clause (in which case the formula is unsatisfiable), which happens when both $C$ and $D$ are empty in the second case.