While I was reading dot and cross products I stumbled upon the well known fact that the dot product yields a scalar and the cross product yields a vector. All I want to know is why. Why does the dot and cross products always yields Scalars and vectors respectively. And also, what led us to define the products the way they are defined?
I think this question is a bit backwards. Asking "why does the dot product always yield a scalar?" is like asking "why is red a color?"
There's no possible answer to this question: red is a color. "Red" is a word which we use to refer to a color, so of course red is a color. There's no reason for it; it's just part of what "red" means.
Likewise, there's no answer to the question "why does the dot product always yield a scalar?". "Dot product" is just the phrase we use to refer to a particular operation which always yields a scalar.
The more important question is "why do we care about these operations of dot and cross product in particular?". The answer is that they are useful, because we can easily compute these operations and learn things about vectors from the computation. For example, two vectors are orthogonal if and only if their dot product is $0$. This gives you a reasonable way to check if two vectors in 700-dimensional space are orthogonal, for example. It's not like you can pull out a protractor in 700-dimensional space and measure angles directly, so this is a very useful computational tool to have!