If one claims to have a scientific bent of mind and he has never wondered about these two at any time in his/her life, I'll doubt 🙂:
For both of these, I'm not talking about the explanation in equations - yes equations help, and actually prove that something is possible. But here we'll be talking about the basic physics part of the phenomenon - what's actually happening out there.
After How Do Objects Float?, let's see if we can understand flying in air (shall tackle floating in air - e.g. clouds - some other time).
For objects that produce thrust-to-weight ratio of greater than $1$ (such as helicopter, birds when they are flapping their wings, dragonfly), there is nothing to explain. I'll be looking at the objects that can fly with thrust-to-weight ratio of less than $1$! Examples are aeroplanes, birds when they are gliding etc. How on earth that should be possible?
This is a magical relation that actually explains everything... let's see how.
On the left hand side we have force - something that we are in dire need of in order to fight gravity. On the right hand side is velocity, and velocity is a component of momentum, and there is nothing holier in physics than conservation of momentum. That is to say, a system or a body tries to maintain its velocity.
What this means in simple terms is that it should be theoretically possible to design a system that can take care of gravity on its own while moving in air. We can give this system, call it aircraft henceforth, an initial velocity and it should be able to generate the required force to fight gravity.
And so, the correct question to ask should be "why do aircrafts need any engines at all for flying?". Well, we'll come to that but first let's build up the idea of creating and using air-resistance and that's where the wings of the aircraft fit in.
This is the first thing that wings do, and engines can't on their own (in any significant way) - they force air to generate resistance.
But we'll have to end the euphoria right here - there is big issue with the formula $F_D \propto v^2$. For a sustained flight, the velocity of the aircraft can be in any direction except along the z-axis, and... the force that we need for fighting gravity has to be strictly in the upward z-direction! So, how to make $v$ in the xy plane generate a force along the z-axis? Turns out, it is actually possible! The solution comes from the fact that the direction of the resistive force from air depends not only upon the velocity, but also on the angle of attack. If we adjust the angle of attack properly, we can get some, if not full, of the resistive force redirected to z-axis upward.
And this is the second job of wings - to redirect some of the air resistance along z-axis. Let's call the total air resistance the aerodynamic force. The following diagram shows in very simplistic manner what's happening. Note that some of the directions are modified in real world, e.g. air velocity after impact - due to Bernoulli's principle and friction etc.
Now that we have some part of the aerodynamic force acting in vertical direction, we should be good to go it seems. We can have sufficiently large initial $v$, and the lift should be able to counter gravity completely. Well, there is one last piece remaining - there is a component of the aerodynamic force that is left out unbalanced - rightly named drag - as it is opposite to the direction of velocity. This force will continuously try to impede velocity, which in turn, will impede lift.
Ok, now we'll summon external power. This is where engine is needed. Engine provides additional force (thrust) for taking care of drag. It's very important to note that this thrust is needed to counter drag and not gravity, and hence it can be, and generally is, quite less than the weight of the aircraft!
So, to summarize, lift is provided by the velocity of the aircraft, and you don't need anything to maintain velocity unless there are external forces. If there are such forces (drag), just take care of those, and the velocity will keep doing its job happily!
Another way to look at it is that wings are always falling onto the air in front of them. This horizontal falling is made to generate some upward resistance that is the lift, while the horizontal component (drag) needs to be taken care of.
The taking-care-of-drag part can be done in many ways, e.g. using thrust from an engine, or by using wind currents (as birds do), or by allowing some free fall and letting drag consume that extra velocity gained.