Why do vector quantities follow triangle law? We usually take two vector quantities, say force, meet the graphical tail to the head of other, and then assuming them to be sides of a triangle, we draw the thrid side, which is the resultant... But why does it work?...this seems quite magical.
 A: I see that existing answers have given reasoning in terms of components. That is ok as far as it goes, but there is a deeper mathematical argument. It is that the mathematical definition of vector is 'a quantity that behaves in the same way as a displacement in space' where by 'behaves' here we mean the mathematical behaviour, such as what happens when you multiply it by a scalar, or what happens when you add vectors together, and what happens when you rotate your coordinate framework. Now, displacements in space add precisely as one arrow added to another: that is simply what we mean by displacement in space. It follows that vectors in general must add like that.
A: For simplicity, consider two forces in two dimensions, $\mathbf{F_1}$ and $\mathbf{F_2}$. These can be decomposed into component form;
$$\mathbf{F_1} = F_{1x}\mathbf{\hat{x}} + F_{1y}\mathbf{\hat{y}}$$
$$\mathbf{F_2} = F_{2x}\mathbf{\hat{x}} + F_{2y}\mathbf{\hat{y}}$$
where $\mathbf{\hat{x}}$ and $\mathbf{\hat{y}}$ are unit vectors in 2D cartesian space. The scalars multiplying them then contain the information about "how much" of the total force is acting in the $x$- and $y$-directions, respectively. When we add the forces, we add their components;
$$ \mathbf{F} = \mathbf{F_1} + \mathbf{F_2} = F_{1x}\mathbf{\hat{x}} + F_{1y}\mathbf{\hat{y}} + F_{2x}\mathbf{\hat{x}} + F_{2y}\mathbf{\hat{y}} = (F_{1x} + F_{2x})\mathbf{\hat{x}} + (F_{1y} + F_{2y})\mathbf{\hat{y}}$$
Geometrically, this corresponds to adding the forces head to tail as you describe, see the figure below. So it's not quite magic, but rather a natural consequence of how vector addition is defined mathematically.

A: The triangle law as you call it is simply just a visualization of what happens when you add vectors. If you have two vectors say $\vec{a} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ and you add them together componentwise you get
\begin{align*}
\vec{c} = \vec{a} + \vec{b} = \begin{pmatrix}1 + 0 \\ 0 + 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}
\end{align*}

The rest then just follows from the linearity of the definition since you can add x and y components individually, they will always form a triangle like that shown in blue.
A: When you add two vectors, let say, $\vec{F}=(F_x,F_y,F_z)$ and $\vec{G}=(G_x,G_y,G_z)$ , you are essentialy adding their components. Those components are just the distances of the proyection on each axis, measured from your point $0$ you chose for reference. E.g, $\vec{F}$ is $F_x$ units of the $x$ axis away from $0$.
The result of the sum is given by
$$
\vec{V}=\vec{F}+\vec{G}=(F_x+G_x , F_y+G_y , F_z+G_z)
$$
You can think of this result as (let's focus just on $x$ axis) moving $G_x$ steps and then moving $F_x$ steps more from there. This generalizes to the rest of components of the vectors, and this is exactly what you are doing when drawing: Moving one vector towards the other one and note where the end is pointing, basically the result of adding components of the first to the second.
In fact, you can even think of your starting point as a vector $\vec{0}=(0 , 0 , 0)$ and describe any other vector as $\vec{U}=\vec{0}+\vec{U}$ since you are just moving in $\vec{U}$ direction from zero.
A: Newton's Principia did not write $\vec F = m \vec a$, or even that $\vec F = \frac{d\vec p}{dt}$. Nor did he write that individual forces add vectorially. The reason he didn't write that is because the concept of vectors postdates Newton's Principia by a couple of centuries.
That individual forces do indeed add vectorially (in the context of Newtonian mechanics) is buried in the eight definitions and the scholium in Newton's Principia that precede Newton's three laws of motion and in the first two corollaries that follow his three laws of motion. The proofs Newton offers for those corollaries is, in my opinion, highly circular. The first two corollaries say that

*

*A body by two forces conjoined will describe the diagonal of a parallelogram in the same time that it would describe the sides, by those two forces apart.
Newton's proof of this parallelogram corollary assumes this corollary is true. His arguments are highly circular.

*And hence is explained the composition of any one direct force $AD$, and of any two oblique forces $AC$ and $CD$, and on the contrary, the resolution of any direct force $AD$ into two oblique forces $AC$ and $CD$; which composition and resolution are abundantly confirmed from mechanics.
Newton's very long-winded proof of this corollary assumes the first corollary is true, and is equally circularly.

Modern physics education instead implicitly assumes what I call Newton's zeroth law of motion, that forces are 3-vectors. One could say that this is magical. Or one could say this is "abundantly confirmed from mechanics".
The history of physics has lots of ideas (e.g., Aristotelian physics, phlogiston theory) that were eventually tossed aside because they were abundantly disconfirmed from mechanics. Despite having been falsified by quantum theory and relativity theory, we still teach and use Newtonian mechanics with its magical concept that forces are 3-vectors because it is approximately valid, to many places of precision, in many realms.
