I am conducting some regressions on economic growth determinants and I am disillusioned. Essentially by choosing what groups of variable to regress I can get the desired significance. What does this tell about whether the explanatory variables influence economic growth?
When you run a regression you are making the assumptions of the Gauss Markov Theorem, one of which is that the error is uncorrelated with all of the independent variables (exogeneity).
If you remove from your regression a variable that is correlated with other explanatory variables then the error term and those explanatory variables will be correlated and the Gauss-Markov Theorem will then be violated. The consequence of this is that the estimates produced by the model will be biased. This particular form of bias is known as Omitted Variable Bias.
A consequence of omitted variable bias is that spurious variables can often be made significant be deleting important variables from the model. In these instances, the fact that they become statistically significant doesn't signify that they suddenly became important, it just means that you are estimating a mis-specified model that violates your own assumptions. So we can't just blindly interpret the hypothesis tests associated with a regression model, we must pay attention to whether the model is well-specified—including the question of whether all of the relevant variables are included.
Last point: your question is "What does this tell about whether the explanatory variables influence economic growth?" The answer is nothing because a regression in itself will not tell you anything about causality, only whether your variables are correlated. To make a causal connection between variables you have to think about how to build causal identification into your empirical strategy from the beginning.