I was able to figure it out. In case anyone (mans in ECE 406) was wondering:
$B \in NP$ means that there exists a non-deterministic polynomial time algorithm for $B$. Let's call that $b(i)$, where $i$ is the input to $B$.
$A$ karp reducing to $B \implies$ that there exists a function $m$ such that $m$ can take an input $i$ to $A$ and map it to some input $m(i)$ for $B$, and if an instance of $i$ is true for $A$ then $m(i)$ is true for B (and same for false case),
Therefore, an algorithm for $A$ can be made as follows:
- Take input $i$ and apply $m$ to yield $m(i)$
- Apply $b$ with input $m(i)$
This yields an output for $A$. Since both $m$ and $b$ are non-deterministic polynomial time, this algorithm is non-deterministic polynomial time. Therefore $A$ must be in NP.