I don't know/remember the historical derivation, but the story you might get in a class goes something like:

Energy can't be created or destroyed, so it follows a continuity equation: du/dt + dq/dx = 0. Roughly, the only way for energy to change in time is by coming from somewhere in space. There are no magic sources/sinks (a source or sink would be a nonzero term on the right).

Then you have Fourier's law/Newton's law of cooling: heat flows proportional to temperature difference, from high to low: q = -du/dx.

Combining these, you get the heat equation: du/dt = d^2 u/dx^2.

Now if you're very fancy, you can find deeper reasons for this, but otherwise if you're in engineering analysis class, just guess that u(t,x)=T(t)X(x). i.e. it cleanly factors along time/space.

But then T'(t)X(x)=X''(x)T(t), so T'(t)/T(t) = X''(x)/X(x). But the left and right are functions of different independent variables, so they must be constant. So you get X''= λX for some lambda. But then from calc1, X is sin/cos.

Likewise T' = λ T so T is e^-λt from calc 1.

Then since it's a linear differential equation, the most general solution (assuming it splits the way we guessed) is a weighted sum of any allowable T(t)X(x), so you get a sum of exponentially decaying (in time) waves (in space).