I was wondering that why does the equivalent resistance actually
increase in a series connection of resistors and why does it actually
decrease in a parallel connection of resistors?
Water analogy: wires are pipes, resistors are pipes with a sponge in it, water is electrical current.
To get through a sponge, the water will flow into holes which eventually end, and then into another hole that the first one connects to, and so on. The more distance it has to go, the more resistance it will see because it has to get through more of the holes along the path. Counteracting this is the surface area of the sponge presented to the flow. The larger the surface, the more "initial holes" you have to flow into, so the water can spread out and flow in parallel through more paths and thus more water gets out the other end in any given time.
Easy demonstration: take a normal household sponge, wet it, and then wring it out. Now hold it flat under your sink faucet and run the water. How long did it take to come out the back? Now wring it out and do it again, but this time hold it so the top edge is facing the water and it has to flow through the entire length of the sponge. How long did that take? Resistance.
So if you place two sponge sections in series, the water has to flow through both, one after the other. This is like when you hold the sponge the long way to the water. So the resistance is twice.
But if you put two sponge sections in parallel, then the water only has to flow through one thickness, and has twice as many holes to start from. This is like when you hold the sponge flat to the water. So the resistance is half.