Could anyone explain how either of these can be proven? I don't see how either of these statements by themselves can be true, much less how to prove them.
First one false when $A=-B=I$. the second one is false when $A=0$ and $B=I$.
They will/may be true for special choice/s of $A,B $ but they are not true in general as shown in the other answer.
For the case that this is true, you'd prove the first statement fir example by taking $x\in N (A+B)$ and showing that this implies also that $x\in N(A)$. That is $(A+B)x=0\implies Ax=0$.