I think it may be useful to distinguish two usages.
In the first usage you are requesting (I use the word request even though the request can be emotionally loaded, for example you may be spurring the other person) the other person to be an "active agent".
Some idiomatic examples are:
-Fatti forza! (Literally "make yourself some strength"), meaning "be strong" but also used when you would say "forge ahead" in English.
-Fatti vivo! (Literally "make yourself alive") meaning something like "keep in contact" but you give the other person the burden to contact you.
But you also have the more straightforward sentences like:
-Fatti dire chi è la spia. (Get him to tell you who the spy is)
On the other hand you can use it to request the other person to be a "passive agent".
-Fatti guardare (let me look at you)
-Fatti dare un consiglio (let me give you a piece of advice)
Note that in this case you use the construction "fatti+verbo (all'infinito presente)"