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As we know, flight vehicles can't keep balance easily if they cannot propel enough air through their blades or wings. We can also see the Falcon 9 at a steeper angle a while before touching down on the landing site.
What are the factors behind its balanced landing?
TL;DR Precise throttling
Falcon 9 first stage booster has a tare weight of around 20 tons (IIRC; keep some buffer though) while landing, so it is heavy enough to not perturb due to minor reflected waves from the landing ground provided that it has shed off enough velocity to avoid hard impact. Also, it is a long slender cylinder and not a typical lift generating body, so any asymmetrical thrust generated by single-engine exhaust during landing is canceled by thrust vectoring.
Apart from that, if any roll is induced (due to grid fin problem or otherwise) is effectively nullified when landing legs deploy (increased moment of inertia) due to conservation of angular momentum. It happened during CRS-16 mission because the grid fins stalled yet you can see the roll is almost canceled before touchdown to the water surface. Not going off the tangent, it's a play of precise throttling and thrust vector control. If any residual thrust remains close to the ground, the booster does tip over as it happens in this video at 0:50 mark and onwards.
Mind you SpaceX didn't reach this stage in a couple of tries. It took them a lot of tests and a lot of failed landing attempts to carefully understand the variables and calculate the optimum set of actions to execute a balanced landing for each mission.
Edit 1- As pointed out by Steve in the comments, the engine-first orientation is largely driven by the center of mass of the booster being towards the lower end. Also, in combination with the above actions, the booster does use cold gas thrusters to maintain a proper attitude towards the landing as can be seen in this video at 0:53 marks. Thanks, Steve!
P.S - Falcon 9 remains inclined to the vertical right till the last minute of the landing because of the fail-safe approach. The booster can autonomously decide not to land at the proposed site if the safety cannot be ensured. This question regarding safety arose after the landing failure of CRS-16 mission (since it was quite near to the coast) and Elon explained it on Twitter.
Just wondering, at what level this question is asked? Stability in rockets is an old problem that goes back to the pre-WWII days of Goddard, Oberth, von Braun, and the like. Goddard tried putting the engine at the top, and it didn't help. Rockets need active control. That means:
1) Sensors that tell you how far off you are from the desired state. For instance, gyroscopes that sense tilt. The Falcon 9 uses inertial sensors and also GPS.
2) A way to drive and tilt the rocket, with engines and other things. Usually engines are gimbaled these days--they can swing around a few degrees, but also vernier engines (small engines used for guidance), and even vanes in the exhaust stream have been used. The Falcon 9 uses gimbaled engines and cold gas thrusters. It uses grid fins to guide it along the descent path, but those aren't very useful when the speed goes to zero.
3) A brain that connects the two in a feedback loop. It takes information from the sensors, determines the deviation from the desired state, and calculates a corrective action. I have no idea what their control software looks like. The workhorse in industry circles is the PID controller, which produces an output from terms that are Proportional to the deviation, the Integral of it (say the average), and the Derivative, or rate of change. I'm sure SpaceX must use something fancier. For instance, it must model the rocket's behavior so it can predict the future effect of a control effort now, rather than just tracking a present-time deviation.
And all of that basically to do what you do intuitively when you walk around with a broom balanced on a finger. If the broom starts to tilt, you can see it and feel it, and you move your finger to catch it.
Here is a video that talks about some of it, but not in the detail that I would have liked.
The main techniques used for balanced landing are throttle control and thrust vector control and cold gas thrusters.
SpaceX uses throttle-controlled Merlin engines that allow them to balance the throttle by lowering it to maintain optimal re-entry speed of the vehicle. Another important feature of the Falcon 9 rockets is their thrust vector control system. This system allows SpaceX to control the angle of the vehicle. Cold gas thrusters help the vehicle in the same task helping the vehicle to maintain overall balance.
