The explanation I always give is as follows:
God never takes away a person's free will.
If God wants to influence a person's choice, He does just that - influences it. He does not force it. He will manipulate external factors so that the decision will be influenced in a certain direction.
Let me give an example: Bill is buying a new car. He has free will to choose whichever model car he would like to buy. Now, suppose for whatever reason, God would like Bill to choose model X. Bill might be influenced in his decision by several factors: (1) Bill's neighbor got model X and loves it. (2) Bill's friend got competing model Y, and Bill doesn't want to look like a copycat. (3) Bill hears of someone who got in a terrible accident due to a car that lacks feature A, which model X is known for. (4) Bill will have had a childhood experience involving model X or similar, which makes him sentimental toward it. Etc. etc.
Because of these factors, Bill chooses model X. He is not forced to, but is influenced in his decision by external factors, which are perhaps manipulated by God to push him toward this decision.
Similarly, "God hardened Paro's heart" means that God provided experiences in Paro's life and surroundings to influence his decision to not allow Israel to leave Egypt. He still, using his free will could have made the right decision, but he willingly chose not to. It might not have been as easy to make the right decision, but he is still responsible for not doing so. Thus, he was justly punished.
Now, why did God need to "harden" Paro's heart? The answer commonly given for this is that God wanted Israel to be slaves in Egypt for a certain amount of time as a prerequisite for properly accepting the Torah. Egypt is described as a "furnace of iron" (I Kings 8:51), in which the metal is rid of imperfections and prepared to be formed properly. Additionally, God felt that the plagues on Egypt were also necessary to show Israel and the rest of the world how powerful and devoted He is. Therefore, God wanted Paro to make the decision to keep Israel in Egypt for their own benefit.