When is it appropriate to apply a transformation? I understand Box-Cox transformations, when to apply which transform (log vs exponential) and on what type of distribution (left vs right skewed). However, what I don't understand is when is it appropriate to consider transformations in the first place? 
 A: This is really to broad, you did not even tell us what is your purpose in modeling, and an answer surely must depend on that!  There is a lot of relevant Q&A here, a short list:


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*Stabilize variance: Why do we stabilize variance?

*Transforming and then using a linear model might be simpler to present to certain audiences than using a GLM:  Why is GLM different than an LM with transformed variable

*Time series modeling is simpler with constant variance:  When to log transform a time series before fitting an ARIMA model

*Many reasons to use a log transform:  In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values?  and When (and why) should you take the log of a distribution (of numbers)?

*Some ideas on transformations and machine learning:  Why aren't power or log transformations taught much in machine learning?

*To satisfy modeling assumptions:  Practically speaking, how do people handle ANOVA when the data doesn't quite meet assumptions?

*Why (or not) use the square-root transform:  Why is the square root transformation recommended for count data?  

*Is it really a good idea to arcsine-transform binomial data? Are ecologists the only ones who didn't know that the arcsine is asinine? or  Can I use a binomial model with logit link function when dealing with continuous proportions?

*Why using Box-Cox transform:  How is the box cox transformation valid? and Box Cox Transforms for regression

*Transformations and principal component analysis:  Why log-transforming the data before performing principal component analysis?

*A good alternative to transform predictor variables in regression models is the use of splines. Examples can be found from Help me fit this non-linear multiple regression that has defied all previous efforts
... and certainly a lot of other reasons. To get some more specific answer you will need to tell us more about your situation! To find more relevant posts here, search this site for "why transform" or to limit the number of hits "why transform score:10.." result (or some variant).
