By order of Ravio, Link heads out to the castle to warn about the magician who has been turning people into paintings. He is stopped by the palace guards who think Link is being ridiculous and is looking for an excuse to see the princess. When it seems like Link won't get into the castle, Impa, the adviser and protector of Zelda, intervenes and lets Link in.
Link waits in the great hall of the castle where he looks at the paintings that represent the events of Link to the Past (if you'd like a recap, you can read the post from part 1 that explains it). Meanwhile, Impa goes before Princess Zelda to pass on Link's warning of the magician. After looking at the paintings for a while, Link is summoned by Impa to come before Princess Zelda.
As Link enters into the throne room, Zelda immediately recognizes him. Zelda believes she's seen Link before, but she does not recognize his name. She does mention that she shared the dream Link had at the beginning of a monster putting Hyrule in darkness. It's important to note that this Link has never met Zelda prior to this encounter. This is almost always the case (with the exceptions of Skyward Sword, the Minish Cap, and the Four Swords Adventure). There is usually some degree of familiarity since the two have been bound by fate for possibly thousands of years. In much the same way Link knows how to use a sword simply by picking one up, Zelda's knowledge of Link comes from her many lifetimes of planning for the days in which Hyrule is in danger.
Zelda gives Link the Pendant of Courage for reasons not yet explained, and she sends Link to an old sage named Sahasrahla. I'm not expecting you to have to remember that name, but his deal is that he was a descendant of the same Sahasrahla from Link to the Past who was the descendent of a sage from Ocarina of Time (the sage of light if I had to guess of course there were six sages in OoT and seven in LttP so maybe there is no correspondence).
I write all of this to say that if you don't know these games that well, Zelda will get captured or kidnapped some time between now and the end of the game. Some critics have said some very strong remarks on how this objectifies Zelda in some manner in that she is frequently kidnapped and doesn't really exhibit much personality. Let's get the record straight on one thing. Link just does things because he's told to do things. I'm not saying he doesn't have some interest in the well-being of Hyrule, but he is not the strategic mind who knows how to stop the forces of evil. Zelda is the master planner. Throughout all of time, Link receives the Triforce of Courage whereas Zelda receives the Triforce of Wisdom. She always delays her capture long enough to allow Link to have the necessary means to defeat whatever evil he must face (usually Ganon). It's also important to note she is the only apparent form of government in Hyrule, which may sound like a foolish means of governance, but it actually makes sense given past attempts at shared power usually lead to the person that isn't Zelda to try and take power over the land in one form or another.
In this sense, it is sort of like a queen in a game of chess. The queen holds the most mobility on the board, but the queen is best used when positioned to thwart enemy mobility. It is the other pieces that take advantage of the queen's position by catching and cornering the enemy pieces to open a way to take the king.
I will also go ahead and say things don't go perfectly according to plan, but the pieces are picked up by someone who has also been planning. We shall see this other planner later. For now, Link must go to Sahasrahla's house in the village.