Suppose you are moving in circle of radius $r$. So there should be centripetal acceleration towards the center. Now you want to decrease the radius of the circle, so someone should apply more centripetal acceleration in order to decrease your radius. But I had read that with the increase in radius, there is increase in centripetal acceleration. What is the correct explanation for without the use of formula? Making it more clear- "My intuition tells me that if I decrease the central force on an object in uniform circular motion, the radius of the circle should increase?"
It all depends on what is kept constant while the radius is changing. If you are keeping the angular speed constant (which is the same as keeping the frequency of revolution or the period constant) then the centripetal acceleration would increase. An example of this would be moving away from the centre of a rotating carousel.
If you are keeping speed constant and increasing the radius then the centripetal acceleration would decrease. An example of this would be driving a curve with a increasing radius (a spiral) at a constant speed.
To understand why, remember that acceleration is the rate of change of velocity. Let's assume for now that we're thinking about circular motion with a constant speed so the acceleration is going to result from a change in direction of the velocity. Consider an object going halfway around the circle. In that time the direction of its motion is going to change by 180 degrees the magnitude of the change of its velocity will be twice the magnitude of its initial velocity. For example if it was originally moving east at 10 m/s and ends up moving west at 10 m/s its velocity has changed by 20 m/s. A bigger acceleration can result in two ways - either from a larger change in velocity, or from the velocity changing in a shorter amount of time.
In the first example (constant angular speed) increasing the radius does not change the time it takes to go half way around. People near the middle or the outer edge of the carousel all take the same time for half a revolution but, the people with a lager radius, near the edge, have to be moving faster. They go around a larger circle in the same amount of time. This means the change in velocity for the people with a larger radius is greater and thus so is their centripetal acceleration.
In the second example the car is driving at a constant speed but as it goes around a curve with a larger radius it will take longer to go half way around so the centripetal acceleration will be smaller because the time for a larger radius is larger.
Here's one formula you might be talking about:
If you want smaller radius $r$, at first glance it seems that you have two options:
Increase the acceleration $a$, or
decrease the speed $v$.
But never be that strict in your thoughts. For example:
I wish to reduce $r$, so I reduce $v$ as well. But if I don't reduce $v$ enough, then I must decrease $a$ a bit as well, to keep the equation true. Now decreasing both $a$ and $v$ still gives the correct decreased $r$.
But if I decrease $v$ too much, then $a$ must increase a bit to make the equation true. Then decreasing $v$ but increasing $a$ gives the correct decreased $r$.
Bottom line: You first of all must know the relationship between the parameters, you are trying to change. Knowing the relationship shows that more parameters than just the two $a$ and $r$ are involved.
And then you must know if any of these parameters are kept constant! Because if they aren't, all parameters might change simultaneously. And when there is more than one parameter, anything can be said and nothing is known for sure because many setups of these parameters can make it work. You can't know then what increases or decreases to give a reduced $r$.
So, be aware of the situation and the conditions. The statements that you refer too, require this knowledge - otherwise they are useless.
For any two situations where an object is in circular motion, assuming the linear velocity is equal in both cases, a larger radius requires less centripetal force to keep that object moving in the circle.
Perhaps an everyday example of this is driving on a highway curve. When you drive around a curve, you feel an "inertial" or "fictitious" centrifugal force, which feels stronger when you have more centripetal acceleration. So if you go around a curve of a small curvature - and therefore small radius - you feel like you are about to be thrown out of the car (the fictitious force), which means you have a stronger centripetal acceleration. However, if you drove at the same speed around a larger-radius curve, then you would not feel as much as much inertial force, and therefore the centripetal acceleration would be smaller.
The key point that is missing from your two contradictory "explanations" is this:
While you are changing the radius, you are not moving in a circle around the original center point. You are moving in a spiral of some kind.
Think about a stone tied to a string being whirled in a circle. If you pull harder on the string to shorten it, the stone starts to spiral inwards, and the string is applying a tangential force to the stone as well as the radial force that causes the centripetal acceleration. The tangential force will increase the speed of the stone around the circle as the radius decreases.
So there are two changes which have opposite effects on the tension in the string. Reducing the radius and keeping everything else the same would reduce the centripetal acceleration, but increasing the speed and keeping everything else the same would increase it.
To find out which effect "wins" you need to do some math, but you said you don't want a "formula".
Also the question says "you" are moving in a circle, but it doesn't say how the centripetal force that keeps you moving in a circle is being applied to you.
So, you haven't fully described what the real-world system is, and you don't want to use the best way (math) to model how it behaves. Therefore, this isn't a complete answer!
Let's use the classic example of a ball revolving around your fingers via a tensed string.
There is a tension in the string because the ball is travelling in a circular motion. The tension force is constantly causing the ball to accelerate toward your fingers. The ball, at all times, wants to travel in a tangentially straight line away from your hand. The tension is constantly preventing this from happening. The tension is related to 3 things; the mass of the ball, the velocity of the ball, and the radius of the string.
If you increase the mass of the ball, the tension will increase to keep it in orbit at its present velocity and radius because more force is require to restrain the greater mass in the existing orbit.
If you increase the radius of orbit and keep the velocity and mass the same, then the orbit is a larger and gentler curve. You could increase the radius so much that the curve would be VERY gentle and theoretically approach the tangential line the ball truly wants to follow. This increased radius and resultant gentler curve causes the restraining tension to DECREASE. If you DECREASE the radius under these same circumstances, the curve is so tight and such a departure from the desired tangential line that the ball wants to follow, then the restraining tension must INCREASE to accomplish this.