I now had some time to think about this again and I think I came up with an explanation. I would gratelygreatly appreciate Ifit if somebody had some comments wheterwhether this is reasonable:
As we know from 2-dimensional CFT where we usually like to work with the variables $z = t+ix$ resp. $\overline{z} = t-ix$ hence holomorphic/antiholomorphic expressions we have:
$$T_{++} \leftrightarrow T_{zz} = T(z) \quad , \quad T_{--} \leftrightarrow T_{--} = T(\overline{z})$$
Sine we can express $T(z)$ resp. $T(\overline{z})$ ad Laurent-series as
$$T(z) = \sum z^{-n-2}L_n $$
where the operators $L_n$ form a Virasoro algebra and generate the conformal transfromations.
Since the physical transformations like dilations/rotations are actually generatetgenerated by the sum/difference of such generators the statement that $(f(x^+)T_{++} - f(x^-)T_{--})$ generates the conformal transformations of our theory seems reasonable.
Furthermore since at point for which x=0 we have
$$x^+ = t + 0 = t- 0 = x^-$$
and therefore
$$(f(x^+)T_{++} - f(x^-)T_{--}) = f(t)T(t) -f(t)T(t) = 0$$ ,$$(f(x^+)T_{++} - f(x^-)T_{--}) = f(t)T(t) -f(t)T(t) = 0,$$
hence the above operator leaves invariant the line $x=0$. Where the implicit statement $T_{++} = T(x^+)$ follows from the one for $T_{zz}$ by Wick rotation