I have read a bit on this topic and have come to the conclusion that this is one of the things in physics we just accept (like it it a "tested" fundamental law (from the beginnings of time etc etc)).
So instead of asking where does it come from, I want to know how conservation of momentum can be connected to other concepts such as newtons laws.
Or a logical thought process to answer how conservation of momentum comes about.
Conservation of momentum follows immediately from Newton's 2nd and 3rd law. For simplicity, I will assume a 2-particle system but the logic can be easily generalised to many particles. For the purpose of the derivation I will also assume that the only force acting on particles is the intermolecular forces between them.
Given two particles (1 and 2) in 3-d, the equations of motion (via Newton's 2nd Law) is given by
\begin{align} \frac{d\vec{p_1}}{dt} = \vec{F_1},\hspace{0.5cm} \frac{d\vec{p_2}}{dt} = \vec{F_2} \end{align} where $\vec{F}_1$ is the force acted on the particle 1 by particle 2 and vice versa for $\vec{F}_{2}$. By Newton's Third Law, we know $\vec{F}_1 = - \vec{F_2} $ so we have \begin{align} \frac{d\vec{p_1}}{dt} + \frac{d\vec{p_2}}{dt} = \vec{F_1} +\vec{F_2} = \vec{F_1} - \vec{F_1} = 0 \end{align} Collecting things up, we get \begin{align} \frac{d}{dt}(\vec{p_1} + \vec{p_2}) = 0 \end{align} meaning the total momentum of the system doesn't change.
