I know that $F=ma$, but why? Why does velocity change in the presence of force? Please answer it based on the fundamental level if possible.
At first, you need to know 'What is force?'
The external cause that causes a stationary object to move or a moving object to stabilize or to change the value and direction of velocity of a moving object is called Force.
As force is the reason of changing velocity or speed, speed changes when force is applied on an object.
Well okay, this is a tricky one.
First recall when is the equation $F=ma$ valid. You may know that it is valid in a special class of frames of reference, known as inertial frames. So now the question is- what is an inertial frame?
If I say that an inertial frame is one, in which the equation $F=ma$ holds, I have told you no new information. So I have to tell you something else.
A frame of reference, in which a particle will keep on moving a straight line trajectory with a constant speed, when all the external influences are taken away from it, is an inertial frame. Note that since I cannot take away all the external influences away practically, so the definition is actually some sort of a limiting case. This is new information- I predict that there exist frames of references, in which particles taken away from external influences move in straight lines.
Now the obvious question is- what if there are external influences present in such a frame of reference? Well, the particles won't follow those special trajectories, that is, their velocities will change. What is a way to quantify the external influences? Certainly, if the rate of change of the velocities is zero, then the external influences are zero. How about equating the rate of change of velocity to the external influence? That is a sensible thing to do. But now we see that for identical external influences, different particles' velocity changes differently. To iron out these differences, we ascribe a quantity called mass to each particle.
So we DEFINE force such that $F=ma$.
Note that the only way to observe whether a force is acting on a particle is to see if the particle is accelerating.
Newton's second law is a way to DEFINE forces.
Physics is the discipline that studies nature, using observations and measurements, and develops mathematical models that can map the measurements and also predict knew measurements. If the predictions are true, the model is validated. If wrong the model is modified or replaced by a better mathematical model.
Mathematics is the discipline that has theories that are self contained. It starts from a few axioms and develops the self consistent formulas . The theories need no validation after completed it is self evident that they are consistent.
Physics uses mathematics as a tool adding extra axioms, laws, postulates and other axiomatic statements so as to pick from the plethora of mathematical solutions those solutions that fit the data and are predictive. These axioms etc come from experimental measurements and observations.
Newton's laws allowed us to use mathematics successfully in order to predict the behavior of matter in our observations. $F=ma$ is one of Newton's laws. thus the answer to a "why" this law is "because" it is necessary to pick up the correct solutions for the data.
Why does speed change in the presence of force
Mathematics allows us to rewrite the second law as $F=dp/dt$ where $p$ is the momentum of a moving mass, ( $p=mv$ $v$ the velocity ) . So the answer to "why" is: "because" it is necessary to assume this axiomatically in order to pick up the correct solutions for the data.
It is the nature of some interactions between bodies that they cause changes in the states of relative motions of those bodies. There isn't a deeper reason for this as far as we know. Interaction is the fundamental concept. Why do things interact? "They just do" - it's part of what they are. There has to be some interaction, I'd think, after all, in order for life and complexity to exist in a universe.
"Force" is a concept created to measure the effect of certain types of interaction. When an interaction is contributing to the change in the momentum (not velocity or acceleration directly), we say that the interaction is "exerting a force" between the objects. The question of why that velocity changes in the presence of force is because momentum is what generates velocity (spatial translation), and so then changes in momentum likewise produce changes in velocity.
Perhaps another way to think of it is this. Objects have positions. Those positions, like anything else, can do only one of two things with time: they can either stay what they are, or change. If they change, that's motion. If they stay where they are, that's stasis. So because there's logically only two possibilities, then it makes sense that one of those two must get realized in a real universe, and so also it should not be a surprise that things in that move in various ways. And so long as a motion is suitably smooth and continuous, you can assign to it an acceleration by pure mathematics.
What is somewhat interesting is that the Universe doesn't seem to require an interaction to change positions of objects, at least not if we define interaction as necessitating "two or more bodies" (I do think it's sensible to talk of "self-interaction", in which case we can talk of an "autokinetic interaction" which translates momentum into changes in position). It only requires interaction to change velocity! Without any interaction with others, they can seemingly keep doing that themselves, forever.
This is not a given! We can easily imagine worlds where that this doesn't happen - in fact, the ancient Greeks thought that that was how the world worked, given that all objects on Earth seem eventually to come to rest, and they imagined that the few celestial motions (like the stars and planets) they could observe required interaction with some sort of external "movers" to make them go, without which they would grind to a screeching halt just like a wheelbarrow without a pusher. The world could have been that way, but it really wasn't.