I am going to answer this in a hurry because the question is on the edge of being closed.
Quantum mechanics isn't just about "wavefunctions", it is also about "observables".
An observable is something like: energy, position, momentum... i.e. it includes all the properties of classical physics.
The wavefunction (or state vector or quantum state) is the thing which gives you the probabilities of observable properties.
There are wavefunctions which correspond to a particular observable having a definite value with 100% probability.
For example, a wavefunction that is a "Dirac delta function" peaked at a point x, might correspond to "particle has position x with probability 100%". (Since you like math details, I will remark that a delta function is not an ordinary function - the one I just mentioned will be "infinity" at x and zero everywhere else - but there are ways to make it a well-defined concept.)
Or, a wavefunction which is a "plane wave" of a specific uniform wavelength, might correspond to "particle has momentum p with probability 100%".
The way to apply wavefunctions boils down to this: You start out with some knowledge of observables - e.g. particles with particular positions or particular momenta. You use the wavefunctions which correspond to those observables having those values, and you evolve the combined wavefunction according to the Schrodinger equation for the physical system in question. You arrive at a new overall wavefunction at a later time, and then you can use that new wavefunction to calculate probabilities for whatever properties you want to know about at that later time.
To get those probabilities, you basically take the wavefunction, express it as a sum (or integral) over wavefunctions that are "100%-probability" wavefunctions for particular values of the property you're interested in, and then you can get the probabilities from the coefficients in the sum/integral.
For example, maybe you have arrived at a wavefunction that is smeared across space, and you want to know about a particle's position. In effect you are re-expressing that wavefunction as a weighted sum of delta functions - and recall that a delta function peaked at x, corresponds to "particle is definitely at point x".
The wavefunction psi(x), a complex-number function which varies from point to point, can be thought of as "integral over all values of x, of psi(x) Dirac-delta(x) dx".
So for example, the delta function at a point x0, contributes with weight psi(x0).
And then the probability rule is that the probability of the actual particle being at point x0, is |psi(x0)|^2.
Alternatively, if you cared about momentum, you would express wavefunction psi as a sum of plane waves of different wavelength. Again, each component has a coefficient, which we might write psi(p) - p for momentum - and the probability that the actual particle has momentum p0 is going to be |psi(p0)|^2.
All these rules have the odd result that you cannot have a wavefunction which corresponds to 100% probability for some particular position and 100% probability for some particular momentum. The definite-momentum wavefunction, a plane wave, is spread out across all positions, and the definite-position wavefunction, a delta function, is a sum over all plane waves if you Fourier-transform it.
That's the uncertainty principle.
What I have described are the bare bones of applied quantum mechanics. You're studying something, like electrons and protons interacting electromagnetically. Perhaps you have a "Hamiltonian" or "Lagrangian" which encodes how they interact in a formula. If this was classical physics, that formula would allow you to start with specific initial conditions - electron here, proton there - and deduce the behavior which follows deterministically from those initial conditions.
But in quantum mechanics, you use that formula differently - you use it to construct a Schrodinger equation for the proton and electron, which tells you how their wavefunctions develop. (Important technical note, this isn't a complete leap in the dark with respect to classical physics, there is a classical equation called the Hamilton-Jacobi equation which anticipates the Schrodinger equation.)
In your question you want to understand "wavefunction collapse". So the most important thing to understand first, is the perspective that maybe wavefunctions aren't real at all. It's position, momentum, and so on, which are the reality. The wavefunctions are "just" a mathematical recipe that happens to give accurate probabilities. The real question would then be, why does this recipe, quantum mechanics, work?
What "wavefunction collapse" refers to, is the part after you have used the Schrodinger equation to evolve a wavefunction through time, when you then calculate probabilities for physical properties. Maybe the result was, 75% probability the particle is at A, 25% probability the particle is at B. Meanwhile, you were also doing an experiment - which the wavefunction calculation was meant to describe - and the particle actually showed up at A.
If you were to then represent the situation "particle at A" by a wavefunction, you would use a Dirac delta function peaked at the point A - or more likely, since you only know A within some experimental error, you'd use a "gaussian" function that is sharply peaked around A.
So you've gone from "wavefunction with some waves at A and some waves at B" to "wavefunction concentrated around A". That is the so-called "collapse of the wavefunction", but the "collapse" here just means that you got some physical data, and started using the appropriate wavefunction. As Rod Vance says in a comment, this is like updating a probability distribution: Flip a coin, before you look you might say "50% heads, 50% tails", you look, it's tails, now you say "100% tails". The probability distribution "collapsed" onto tails.
Back to your question: Quantum mechanics says nothing about "how does a wavefunction collapse", because it says nothing about whether a wavefunction exists in the first place. The wavefunction is first of all part of a calculation, that gives you the odds of the particle arriving at A or B, given some initial condition.
The real question is, what is really going on? The question, what causes the wavefunction to collapse, already contains the assumption that wavefunctions are physical things and that they have collapsed by the time e.g. the particle is at a definite place. So it's a question that is appropriate for a particular attempt to get beyond applied quantum mechanics to some new physical theory or some understanding more fundamental than "do these calculations and it works"... namely, the path where you hypothesize that wavefunctions are real things.
Incidentally, I should mention decoherence. This is a thing that the Schrodinger equation can cause a wavefunction to do, but it is not the same as collapse. It doesn't take a particle whose wavefunction is at A and at B, and produce a wavefunction just at A (for example). What it does, is to take the combined wavefunction for e.g. a particle and a physical "pointer" with two values, the A-value and the B-value, and produces a wavefunction with a peak at "position A and pointer value A" and another peak at "position B and pointer value B". This means that decohering interactions are good for measurement, but the wavefunction evolution in itself still doesn't produce a single definite outcome, you still have to apply the probability rule to the decohered wavefunction, which in the case I just described is still a "superposition" over two possible outcomes.
I have written this informal tutorial about quantum mechanics because the ultimate fact is that no-one yet knows what is really going on. There are people who don't care about anything beyond applying the quantum formulas; there are people who somehow believe that reality isn't there before observation; there are people who try to make a classically objective theory just out of wavefunctions; and there are people who try to make a theory in some other way.
And meanwhile, people continue to develop new theories within the quantum framework, all the way up to string theory (which is this whole apparatus of wavefunctions and uncertainty principle and observables, applied to vibrating interacting "strings"). This progress within the quantum framework has gone very far and become very sophisticated. But, though lots of people have ideas, and lots of people will tell you that they already know the answer, the primordial question of what lies within or beyond quantum mechanics remains unanswered.