What does this expression from gradient descent mean?
After reading up on multivariate calculus, I have understood what the above equation means. It is a simplified version of the multivariate chain rule - a rule used to differentiate multivariate functions. Here is what it looks like:
$$\cfrac{dC}{dt}≈\cfrac{∂C}{∂v_1}  \cfrac{dv_1}{dt}+\cfrac{∂C}{∂v_2}  \cfrac{dv_2}{dt}$$

Time applies here because we are studying how the cost changes over time/over the course of training. Multiplying each side by $dt$ gives:
$$dC≈\cfrac{∂C}{∂v_1}  dv_1+\cfrac{∂C}{∂v_2}  dv_2$$

And since the differential operator $d$ means a 'small change', we can replace this with delta to represent the same thing, thus arriving at the equation from the question:
$$\Delta C≈\cfrac{∂C}{∂v_1}  \Delta v_1+\cfrac{∂C}{∂v_2}  \Delta v_2$$
To summarise, this equation is a simplified version of the multivariate chain rule, which although means the same as the original, does a better job in communicating that a 'small change' in $v_1$ and $v_2$ results in a 'small change' in $C$.