"I believe the interpretation of the p-value is that it is the probability of seeing your sample's test statistic under the null hypothesis."

No. It is the probability to see your sample's test statistic *or something that is even less in line with the null hypothesis* ($H_0$) under the $H_0$, which I write as $P_0\{T>t\}$, where $T$ is the test statistic and $t$ is its observed value, assuming here that a large value of $T$ provides evidence against $H_0$ (the argument can as well be made for the $\{T<t\}$ or the two-sided case).

If you have, say, $p=0.06$ in one test $T_1$ with result $t_1$ and $p=0.6$ in the next ($T_2, t_2$; let's assume they were done on independent observations), if you multiply these two, what you get is the probability of $\{T_1>t_1\} \cap \{T_2>t_2\}$, i.e., the probability that $T_1$ *and* $T_2$ are too large under the $H_0$. This is of course less likely than having at least one of them too large. But there are cases with at least one of them too large that count at least as strongly against the $H_0$, such as having $T_1$ extremely large even if $T_2$ doesn't indicate problems with the $H_0$, so the event $\{T_1>t_1\} \cap \{T_2>t_2\}$, of which you get the probability by multiplying the p-values, does not cover all possibilities to observe *something that is even less in line with the $H_0$* than what you observed, and is therefore smaller than a valid "combined" p-value would need to be. 

In my example above, surely after having observed $t_1$ with $P\{T_1>t_1\}=0.06$, observing $t_2$ with $P\{T_2>t_2\}=0.6$ doesn't make the overall result indicate any stronger against the $H_0$ (as multiplying the p-values would suggest), because observing something with $P\{T_2>t_2\}=0.6$ is perfectly reasonable under $H_0$; however observing $T_1$ even larger than $t_1$ would arguably count stronger against $H_0$ even with observing a smaller $T_2$.

The problem with combining p-values from more than one test is that if you only have a one-dimensional test statistic, as long as this statistic is suitably defined, it is clear how you can find all possible outcomes that are less in line with $H_0$ than your observation (depending on the test statistic either by looking at all larger, or all smaller values, or combining the two sides). However, with two or more values of the test statistic, in the higher dimensional space of possible outcomes it is much more difficult to define what "less in line with $H_0$ actually means. One possibility to play it safe is to look at $P(\{T_1>t_1\}\cup\{T_2>t_2\})$, the probability that at least one of $T_1$ and $T_2$ is too large. This for sure covers all possibilities that the pair $(T_1,T_2)$ is less in line with $H_0$ than the observations $(t_1,t_2)$. It actually covers too much and is therefore a very conservative choice, but at least valid. It may however be seen as useless, because its probability will always be bigger than $P\{T_1>t_1\}$, so this won't allow you to find a significance based on  $(T_1,T_2)$ if you don't find one based on $T_1$ alone.