What does $ d\tan(x) = \sec^2(x)\,dx $ mean?

It's really just shorthand for writing out the full variable replacement. Instead of writing: $$\int \sec^2(x)dx= \int \left(\frac{d}{dx} \tan(x)\right)dx= \tan(x)+C$$ You can just write: $$\int \sec^2(x)dx= \int d\tan(x)$$ Which is just like writing:

$$\int dy = y+C \ \to \ \int d\tan(x) = \tan(x)+C$$

As your example in the comments suggests this works for more complex integrals as well:

$$\int \sec^2(x)\tan(x)dx = \int \tan(x)\ d\tan(x) = \frac{\tan^2(x)}{2}+C$$ Much like: $$\int y dy = \frac{y^2}{2}+C $$

Differentials measure the tendency of a function to change given an infinitesimally small change of the dependent variable. The expression $d\tan x=\sec^2 xdx$ means that the difference $\tan x_1-\tan x_0$ is approximately $\sec^2x_0\cdot(x_1-x_0)$ when $x_1$ is near $x_0$.

In other words, if we reduce to infinitesimal changes of $x$, we are saying that the slope of the function $\tan x$ at a point $x_0$ is $\sec^2x_0$.

As others have already noted, the value of expressions like this in computing integrals is that it is often easier to see an antiderivative having changed variables. In some sense, we are allowing $\tan x$ to become our new independent variable in the integration.