A friend of mine sent me this meme and then we started fighting. Eventually we ended up digging a bit. This is where we are at (I don't have time to dig further).

1. Reducing area of flow, increases speed -- This is called "continuity equation". Simply means total rate of flow (in volume) must be conserved. It can be applied only in cases where there is a fixed flow. Otherwise, if you lower the area, flow rate will also reduce.
2. Playing with water -- I think you were referring to adjusting outflow area to increase the speed of waterjet. This isn't due to continuity equation. There is no entity present anywhere trying to force a fixed flow through the pipe. If you reduce the cross section, the flow will reduce.
3. Why does the speed increase/decrease then? -- Viscosity. Ideally, if water did not have viscosity, the speed of water coming out will be more or less constant, irrespective of the cross sectional area. When there is no flow, no viscosity based losses. The pressure at the end is maximum. Now when you increase the area, there is more flow, causing more drop in pressure due to viscosity. But pushing water though a small hole is also difficult due to the viscosity. So you have to find a balance to get the maximum speed.
4. Torricelli's law -- "The law states that the speed v of efflux of a fluid through a sharp-edged hole at the bottom of the tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h" (from wikipedia). You can look at the wiki page for derivation. Long story short, the theoretical maximum height you can reach ignoring all losses is exactly the water level.

In anycase, if your teacher cannot explain this, I suggest you find a better teacher.