Intuitively, for the homotopy to be generic has the same meaning as for the Morse function itself to be generic: the set of points where it fails to be a submersion is as simple as possible. Practically, this means that for all but a finite number of times (values of the $t$ parameter) the function $H(x,t)$ is a Morse function of the variable $x$ and the Morse singularities trace out smooth curves transverse to the $x$ direction; and for each of those exceptional times $t$ the function $H(x,t)$ is Morse at all values of $x$ except for a single value $x$ (I'm assuming compactness here) where it undergoes one of a certain class of very special singularities obtained by collapsing two Morse singularities, called "birth-death" singularities. Just as with Morse singularities themselves, which are described locally by specific functions of $x$ in some coordinate system, birth-death singularities are described locally by specific functions of $x,t$ in some coordinate system.
The function $H(x,t) = x^3 - tx$ at $t=0$ is an example of a birth-death singularity in one dimension: for $t>0$ there are no critical points; and for $t<0$ there is one maximum and one minimum.