When we walk, we apply a force $F$ on the earth at an angle $\theta$ . Thus, by Newton's third law, $$ F_{\text{me, earth}}=-F_{\text{earth,me}}$$ Therefore, the earth exerts a reaction force on us, the horizontal component of which enables us to walk. The force I apply also makes the earth move an extremely tiny bit but not too much because of its gigantic mass.
Is this explanation correct? What role does friction play in all of this?
Your answer is correct. If you don't have friction you cannot apply a horizontal component to the force.
Let's focus on the foot.
If you are standing still of course there is no friction, every force is just acting on the vertical. But let's think of what happens when you start to walk. If you think about it you're just, in a way, "pushing the floor" in the opposite direction of your walking. Thus the friction force acts in the same direction of your walking.
In the picture above you can see the forces that the Earth exerts on your foot:
Think about walking on a very slippery plate of ice. You can't walk because your foot won't stay put. Your foot would go backwards, as your foot is exerting a force in that direction. When the friction comes to play it opposes that force and your foot can stay still.
The moving-forward movement can finally take place thanks to BOTH friction force and the work of your knee and hip articulation.
The force responsible for allowing us to walk is static friction, but the way it plays a role can be somewhat unintuitive. I want to make it clear I am not an expert in biomechanics, physiology, or sports science. My discussion will be a highly simplified overview of what role friction plays in walking.
First, let's suppose we were trying to walk on a flat surface with a ludicrously small coefficient of friction (static and kinetic). Maybe it's slippery ice with a layer of water or oil.
For the sake of the discussion, we approximate the friction forces as zero $\vec{f}\approx 0$. But if this is the case, then there are no external horizontal forces acting on you, which means your center of mass must not have any horizontal acceleration. If you start with zero velocity, your center of mass will remain at the same location!
So what happens when you try to walk on such a surface? Imagine one foot in front, one foot at the back. When you lift your back foot and move to forward, a portion of your mass (the leg part) moves forward. But because your center of mass remains at the same location, the rest of your body (including your front leg) must move backwards. In the end, you only change positions of your legs without changing your location whatsoever.
Another way to think about it is through Newton's third law. Absent of external forces, if you try to pull one part of your body in one direction, the rest of your body will be pulled in the opposite direction. Hence, you "won't be able to win," so to speak, and achieve overall translational displacement.
Now contrast this to the case where friction holds your front foot in place as you move forward (so the coefficient of static friction is sufficiently large). As you can intuit from your everyday life experience, the static friction will keep your front foot in place. Somehow this is responsible for your overall motion forward.
Now there are a few interesting things to note:
These certainly seem paradoxical.
To understand things more clearly, we can take a closer look and take a few things apart. Let us zoom in on my drawing containing the "stick figure with brick shoes." I will depict horizontal forces only.
When one foot is on the ground and you are in your swing phase, your body is applying a force onto your grounded foot pulling it backwards. By Newton's third law there is a reaction force from your foot onto your body pulling the body forwards.
In the absence of friction, your foot and body will just switch which one is in front without any change in your center of mass. However, because there is static friction onto your grounded foot canceling out the force your body is applying, the grounded foot stays in place while your body is able to "swing" itself forward. Once your body gets in front of your grounded foot, you managed to make one complete step.
Related questions (some answers there might be relevant as well):
Its the frictional force. When you walk your feet push the ground backwards and according to Newton's third law, the ground pushes you forward.
The ability to overcome the gravitational force and atmospheric force which requires energy.
Can you walk without a reaction force in space??
So reaction force is necessary which is provided by one leg and the other leg alternately during walking.
That's how human evolved !!!
