Hi,
"every right coset of H in G is a left coset of H in G" means that there exists some b in G such that aH=Hb. so aHb^-1 = H and because the identity element e belongs to H we have ab^-1 in H. Since H is a group, we also have (ab^-1)^-1 = ba^-1 in H. In other words, Hba^-1 = H.
Now we have aHa^-1 = Hba^-1 = H.