"Dat is een prijs die ik me kan veroorloven."

Translation: That is a price that I can afford.

[SrMarien]

The pronunciation of 'veroorloven' is sooo poor on this one.

http://forvo.com/word/veroorloven/#nl

Second one on forvo is the best one IMO

[MartinFs00]

What is the difference between die and dat in this sentence? And what is "me" doing here? Does it have to be veroorloof me even if u say ik before?

[Paadrijg]

It is "de prijs", therefore when referring back to it you need to use "die". The "dat" at the beginning of the sentence is simply to specify which price you are talking about, in this context.
"Veroorloven" is a reflexive verb, so it requires the additional pronoun (i.e. "me", "zich", "ons" etc. depending on who it involves)

[MartinFs00]

Thank you :D