Translation:You can meet somebody that you like.
In italian "puoi incontrare qualcuno che gli piaci" is senseless. "Puoi incontrare qualcuno che gli piace", on the contrary, has sense but not the one you think: it means "you can meet somebody that he likes". The correct translation of "puoi incontrare qualcuno che ti piace", is the one given by duolingo, that is "you can meet somebody that you like". "You can meet someone who likes you", that is your phrase, has a very difficult italian translation: It is "puoi incontrare qualcuno a cui tu piaccia".
"Piaccia" is part of a verbal form called "congiuntivo presente". It comes from the Latin, therefore is a peculiarity of romance languages (italian, spanish, french, portoguese ecc.) and has no direct corrispondances with any english verbal form. Explaining in which cases the congiuntivo tense is to be used is a hard task, anyway it is possible to say that 9 out of 10 times, when a verb is preceded by the conjugation "che", that verb is to be in the congiuntivo tense, that comprehends congiuntivo presente, imperfetto, passato and trapassato. The conjugation of "piacere" at the congiuntivo presente is "io piaccia, tu piaccia, egli piaccia, noi piacciamo, voi piacciate, essi piacciano". If you are interested in understanding how much the congiuntivo is difficult to learn (above all in italian), you should look for the conjugation of "piacere" in the congiuntivo imperfetto, passato and trapassato. Eventually, for what concernes the rest of the phrase "puoi incontrare qualcuno a cui tu piaccia", "a cui tu" is an expression that has to be followed by a verb in the congiuntivo, and that, in this case, is the only way to express the fact that the "person you can meet" likes you.
Just to add to Riky's explanation the il congiuntivo presente is, in English, the present subjunctive, one of 4 subjunctive verb forms. it contrasts with the indicative mood by introducing an element of subjectivity and expresses opinion emotion or uncertainty. Since we don't have this verb form In English it is difficult for us to come to terms with it.