The $\mu$ is the ‘drift’ term which represents the average change in the value, ie whether it tends up or down. The $\sigma$ is the ‘volatility’ term which describes the severity of the variation, in fact it is often taken to be the square root of the variance.
Imagine a stock market graph rapidly going up and down in short time intervals, but with a general trend going either up or down over a longer time interval. Then the slope of this general trend is the drift because it ‘drifts’ up or down at this rate, while Brownian motion has expectation zero and so the drift encapsulates all of the expected increase or decrease. Again picturing this stock price graph, the size of the small bumps in short times is measured by the ‘volatility’, ie how volatile is it? This is in general an independent question to the general trend (unless you specify volatility to change over time or with $X(t)$ of course).
So you can see that a change in $X(t)$, ie $dX(t)$, is the slope $\mu$ of the trend times the change in time $dt$, plus the little ups and downs, which is described by a Brownian motion $B(t)$ and scaled by the volatility $\sigma$.
I agree that the concept of $dB(t)$ when $B(t)$ is a very strange creature, but this then makes it all the more obvious why this stochastic calculus forms it’s own branch of maths, it’s a much more abstracted topic.
The idea for what $dB$ is comes from how we define an integral of a stochastic/random process. Much like a Riemann sum, we take the sum of values of the process in intervals $\left[B_{t_i}, B_{t_{i+1}}\right)$, weighted by the length of the interval. The ‘mesh’ is the maximum interval length and as this tends to zero, we reach a quantity which is both well-defined and converges in probability. To my knowledge, lots of textbooks which cover stochastic calculus show this definition of the stochastic integral.
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