Seems like it curved as it went down, I did not expect to see that but, thinking about it, no reason it should just go straight down.
At the scale of a well, the steel drillstring is more like cooked spaghetti. You don't "push" down, you use the weight of fluid (drilling mud) for pressure. The drillstring is only meant to transmit torque. It's easy for it to bend over long distances. (You can even turn things entirely around and use fluid pressure to turn the bit, in which case the drillstring is basically a steel hose that's stored coiled up.)
Is the drilling mud pumped down to create that pressure then, or is it naturally present? Or one pumps water down to turn the grit into a slurry?
You need a steady gradient of pressure. The pressure you need at the bottom of the hole will fracture the rock in the shallow section. Pumps add a constant pressure, so there's no way to get the right pressures at each depth with a pump.
Instead, drilling exploits gravity to do the hard work. The drilling mud is very dense (almost as dense as the rock) so that the pressure at the bottom of the borehole from the weight of the mud is just a bit less than the pressure from the weight of the rocks.
As far as how they do it, it's conceptually simple (these days there are "fancier" options with steerable drillbits). You put a tool down the borehole that you can expand and lock in place at the desired depth. When it's expanded, it forms a ramp towards one side of the wellbore. You then lower the drillstring, and when the bit hits the ramp, it's deflected in the direction you want it to go.