I guess the first column is the coefficient value, the second one is
standard error, and thereafter are $t$ and $p$ values.
Before: dummy is close to 0. This means that flat/progressive tax
system has little effect (no relationship to causality ;-) after
controlling for the continuous variable (call it $x$).
After: dummy is -0.043 and significant. In case the dummy = 0, you
just ignore the interaction (it is always zero) and you just look at
the $x$ estimate. For observations where dummy is 1, you have to add
the interaction term and the $x$ estimate, and you get $-0.043 + 0.078
+ 0.030 = 0.065$. If $x = -1$ you get the tax system effect analogously
$-0.043 - 0.078 - 0.030 = -0.151$.
The fact that insignificant dummy turns negative when you add
interaction has rather simple intuition:
Before, you assume that both cases (dummy = 0 and dummy = 1) have a
common slope ($x$ coefficient) but different intercepts
Afterwards, you allow both slopes and intercepts to differ.
Obviously, changing slope will most likely change intercept too
unless the slope line "turns around" $x=0$.