I'm trying to understand the following claim:
if the $t$-statistic is greater than zero, it indicates that the variable is explosive... but does that mean it has unit root?
In the context of Dickey Fuller test.
First of all it could be useful to read a bit about the unit root problem (you may start from the hypothesis section). So the nature of the explosiveness (exponential growth) is what matters. Roughly the growth could be explained either by deterministic part (for example linear trend) or by random walk with drift. Dickey Fuller (not the best unit root tests choice though, if you are familiar with $R$ I would suggest to go for library(urca)
Zivot and Andrews test ?ur.za
, since this one also counts for possible structural breaks) is designed to distinguish between the too, thus $t$ statistic will help to decide if the nature of explosive behavior is deterministic or not. But bear in mind that DF test is very sensitive to deviations from the assumptions it was build on, even ADF would be more robust to apply in practice.