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We know the formula, $$m = m_0/(1-v^2/c^2)^{1/2}$$
since $v$ is usually way less than $c$, ($v^2/c^2$) is less than one which indicates a body's mass increases when it has a speed. Can someone please help me get an intuition of why this happens in a basic level without using any high-level maths or theories?
Let's rewrite this using $E=mc^2$ to show how the energy increases with velocity \begin{equation} E = m c^2 = \frac{m_0 c^2}{\sqrt{1-\frac{v^2}{c^2}}} \end{equation} Hopefully it is intuitive that the energy of a particle should increase with velocity.
If you know about Taylor series, you may know that when $v \ll c$, we can approximate the square root in the following way: \begin{eqnarray} E &=& m_0 c^2 \left(1 + \frac{1}{2} \frac{v^2}{c^2} + \cdots\right) \\ &=& m_0 c^2 + \frac{1}{2} m_0 v^2 + \cdots \end{eqnarray} where the $+\cdots$ refers to terms which have additional powers of $v^2/c^2$, and therefore are small if $v \ll c$. After doing this approximation, we see that the relativistic formula for the energy $E$ reduces to a constant $m_0 c^2$ (which does not directly affect any observables because you can always add a constant to the energy), and the non-relativistic expression for the kinetic energy, $\frac{1}{2} m_0 v^2$.
Therefore, we can think of the formula for the relativistic mass that you gave, as a formula for the relativistic energy, using $E=mc^2$. The relativistic energy generalizes the non-relativistic idea of kinetic energy to the situation when $v/c$ is not much less than $1$. ($v/c$ will always be less than one for a massive particle, but as $v$ gets close to $c$ one cannot ignore relativistic effects).
Finally, note that it is not very common to use the relativistic mass these days. Most people use the word "mass" in the context of special relativity to refer to the invariant mass $m_0$.
I suggest you think of it in the following way...
Newton's laws tell us that when you apply a force to an object, the resulting acceleration is inversely proportional to the mass of the object. So you can think of mass as being a property that determines how much an object accelerates when a given force is applied to it.
Now you need to consider something called the relativistic law of velocity addition. What that tells is that if you add two speeds, A and B, say, the resulting speed isn't just the sum A+B. Instead, the resulting speed is always less than A+B by some difference. When A and B are very low speeds, the difference is so tiny as to be undetectable, but the difference gets bigger and bigger as the speeds involved get closer to the speed of light. So if, say, you are travelling at 0.9c in one direction, and I am travelling at 0.9c in the other, our relative speed is not 1.8c but 0.9945c.
Now imagine that you are at a bowling alley and you bowl a ball towards the pins at 3m/s. From your perspective, you have applied a given force to the ball, and you have accelerated it to 3m/s in accordance with Newton's law. However, if your bowling is viewed from the frame of a muon passing at 0.9c, the speed of the ball after you have bowled it will appear to have increased by less than 3m/s- in fact, it will have increased by only around 1m/s in the muons frame, owing to the law of relativistic velocity addition.
So, in your frame, applying a force to the ball increases its speed by 3m/s while in the muon's frame the same force has increased the ball's speed only by 1m/s, so it is as if the mass of the ball has increased by a factor of three from the muon's perspective, if you assume Newton's law still applies.
In summary, mass seems to increase with speed because of the law of velocity addition.
Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies". The incompatibility of Newtonian mechanics with Maxwell's equations of electromagnetism and, experimentally, the Michelson-Morley null result (and subsequent similar experiments) demonstrated that the historically hypothesized luminiferous aether did not exist. This led to Einstein's development of special relativity, which corrects mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as relativistic velocities). Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible
You say "we know the formula"
$$m = m_0/(1-v^2/c^2)^{1/2}$$
The formula is part of special relativity which Einstein proposed in order to reconcile data with a theoretical model, the one of special relativity which instead of using three dimensional vectors uses four vectors.
Four vectors were shown to be the proper model for spacetime, allowing to describe all kinematics and energetic behaviors of particles, from elementary to complex one for all velocities, tying up energy and mass in the formulas. At the limit of low velocities the every day Newtonian kinematics are recovered.
So the answer to the question "why special relativity is supposed to be true" the only answer is "because it fits the data.
So to your "why this happens"
is "because that is what fits the data".
P.S."because that is what fits the data" should be qualified ,because for very large masses special relativity becomes the lower energy limit for General Relativity, but that is an other story still being studied as far as energies go.
I mean, "why" is a fun question because you're eventually destined to just get an "because the universe says so" answer, but I'll try and give the shortest, most intuitive explanations I can think of down a few levels. Feel free to stop whenever the answer feels satisfying. :)
1. Your question: "Why does the relativistic mass of a body increase with its speed?"
My response question: "Well, what's mass?"
In the most basic sense, mass is nothing more, nothing less, than "The thing that makes you push harder when you accelerate something."
If I have two stationary objects and one has twice the mass of the other, and I push on both the same non-ridiculous amount, the bigger one will accelerate half as much.
When the thing is slow, the amount of "push" it takes to accelerate something doesn't change with how fast the thing is going. See Newton's Second Law.
When the thing is fast, it takes more extra "push" to get a thing going. The faster it goes, the more extra "push" it takes. If you want to accelerate something with mass up to the speed of light, it takes an infinite amount of force. Since mass is how we quantify how hard it is to push something, that means we think of it as having a mass that approaches infinity as its velocity approaches the speed of light.
2. But why does it take an infinite amount of push to get a massive object moving the speed of light?
That's a bit less intuitive, but we can remember that force causes a change in momentum with respect to time.
$$ F = \frac {dp}{dt}$$
And when you have something going really fast, the faster it goes, the more momentum it gains.
$$p = \gamma m_0 v = \frac {m_0 v}{\sqrt{1-v^2/c^2}}$$
3. But why does a particle gain more momentum the faster it goes?
Because a single photon of light has the SAME momentum no matter what perspective I'm looking at it from. It also has the same speed. Speed is change in distance over change in time, so for that to be true, distance and time must vary based on your inertial frame (perspective). Specifically, if something is moving relative to me, a unit of time on its internal clock will appear to take longer than in mine. And a unit of distance on its internal ruler will appear shorter than in mine.
We're entering "sacrifice accuracy for intuition" territory here since if we're actually accelerating a relativistic particle, special relativity doesn't really apply, buuuuut....
Let's pretend that I apply some constant impulse (relative to me) over 1 second to this particle. Relative to the particle, the faster it goes, the shorter that 1 second becomes from its perspective. As it approaches the speed of light, I'll need an infinite impulse (infinite force over finite time, or finite force over infinite time or both), giving it an infinite momentum, in order to get it to reach c.
4. But why does a photon have the same speed no matter what my reference frame is?
Because Electric fields are generated by charges and Magnetic fields are induced by moving charges in a very specific way. And the way those fields cause the universe to behave actually agree even though the fields themselves are different depending on your perspective (for example, in the lab frame a charge has an E and a B field since it's moving, but from the moving charge's frame, there's no B field). Translation: As Jensen Paull said above, Maxwell's equations hold in inertial frames.
5. But why do Maxwell's equations hold in inertial frames?
Because the universe says so.
The abrupt answer to your question is: because you defined it that way.
The arrogant answer: no physicist uses relativistic mass.
The useful answer: relativistic mass is really not a useful quantity, it is just a way of writing $E/c^2$. But this statement actually contains an answer to the thing which I believe is at the core of your puzzlement. Once a body approach the speed of light, you need to increase the body's (kinetic) energy by ever larger amounts to achieve ever decreasing speed ups. This is the effect of the factor $\frac{1}{\sqrt{1 - v^2/c^2}}$ in your expression, rewritten in terms of energy $E=\frac{m}{\sqrt{1 - v^2/c^2}}$ (where $m$ doesn't depend on $v$). So you need to do more work to gain speed, and this may or not agree with your intuition that this corresponds to the body becoming more "massive" (or more "inert")
So why is relativistic mass not a useful quantity? Because its meaning is entirely subsumed by energy. Mass is nowadays commonly used to refer to "rest mass" and that is a useful quantity which has the property that $m^2 c^4 = E^2 - p^2 c^2$ which constrains the possible values of $E$ and $p$ for a body of (rest) mass $m$.
An aside: the last expression also works for massless particles, which cannot come to rest. So calling "mass" the "rest mass" in that equation would actually be confusing.
