Do the statistical testing results of a transformed variable apply to the original variable? Suppose a variable is transformed, and a statistical test is applied to the transformed variable. Do the results of the test (specifically, p-value) apply to the original variable?
For example, suppose I have a variable which does not appear normal - but it does appear normal after log-transformation. So I apply log to the variable, and perform a statistical test (for example a t-test, or an Anova with a post-hoc test).
If the statistical test gives a pvalue < 0.05 can I declare that the original variable is statistically significant between conditions?
If the answer is yes, then why so?
If the answer is no, then what is the use of data transformation?
 A: 
If the statistical test gives a $p$ value < 0.05 can I declare that the original variable is statistically significant between conditions?

No.
Assuming you are using a linear regression model, and you log-transformed the predictor (although the same applies when you log-transformed the response): the $p$ value tests the null hypothesis of no linear association between predictor and response. With a $p$ value < 0.05 (assuming an $\alpha$ level of 0.05) you would reject the null hypothesis of no linear association. If you have log-transformed the original predictor variable, then you would thus reject the null hypothesis of no linear association between the log-transformed predictor and the response. This tells us little if anything about the statistical significance of a linear association between the original predictor and the response.
The answer to your question would be yes only for linear transformations of the predictor (and/or response). Then you would obtain the exact same $p$ value before and after transformations.
Transformation of the predictor(s) can be helpful, for example, if you do not expect a linear association between the original predictor and response, but a different shape of association. This is the underlying idea of generalized additive models. Note that linear regression with normally distributed errors does not assume the response variable to be normally distributed; only the residuals. Transforming the response might complicate interpretability of the resulting model, and might not be necessary.
