First, consider the edge case where the sail is acting as a bag when you're sailing downwind. As the boatspeed approaches the true windspeed, the apparent windspeed falls to 0 and the sail will luff. In this specific case, the boat can not go faster than the wind.
Now consider the boat cutting across the wind at a 90 angle. When the boat starts moving, the wind comes 90 degrees off the bow. As the boat increases speed, the apparent wind shifts closer to the bow. Apparent wind is just vector addition of true wind and boat wind. If the boat achieves the same speed as the true wind, then the apparent wind is sqrt(2) ~ 1.4x faster than the true wind. More wind means more power, so with that additional wind, it can go faster. Continuing the example, as the apparent wind increases, it appears closer and closer to the bow. Eventually the sail will stall and produce less lift. This is the point where the boat will go no faster.
The slowest point of sail is directly downwind. In a race, it is often much faster to gybe back and forth rather than ever go directly downwind. When a boat goes directly downwind, their boat speed cancels out the true wind. In the strangest case, if a high performance boat going faster than the speed of wind (say, on a broadreach) goes directly downwind, the apparent wind will appear to be coming head on. They've effectively gone 'into irons', yet they're facing 180 degrees off true wind.
If you ever get the chance, you should see the SailGP boats race. Their sails are almost always hauled fully in, even downwind. The other thing is that they gybe downwind because to go directly downwind would be to stall. In effect, these boats can achieve multiple times the true wind speed, but so long as they aren't pointed directly into, nor directly away from the wind.
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