Quick answer
probar = to taste, to test, to try something out, to prove.
probarse = to try on clothes, shoes or glasses — on your own body.
Anything you wear takes the pronoun. Everything else does not.
Side by side
Prueba el vino. / Pruébate la chaqueta.Tasting the wine versus putting the jacket on.
¿Me lo puedo probar?Can I try it on? — the sentence you actually need in a fitting room.
Probé a llamarla.Probar a plus infinitive means to give something a go: I tried calling her.
Memory trick
If the thing ends up on your body, use the pronoun. Food, methods and machines stay pronoun-free.
Typical mistakes
¿Puedo probar este pantalón antes de comprarlo?
¿Me puedo probar este pantalón antes de comprarlo?
Clothes go on your body, so the verb needs the pronoun. Without it you sound like you want to taste the trousers.
Me probé la tarta de tu madre.
Probé la tarta de tu madre.
Food takes plain probar. The pronoun belongs to clothes only.
Check yourself
Fill in: "¿Puedo ___ estas botas en la talla 40?"