Here is a translation of Strebel's definition.
First of all, a quadratic differential on a complex manifold $M$ (holomorphic or not) is a smooth section of the symmetric square $S^2(T^{*(1,0)}M)$ of the holomorphic cotangent bundle; if $M$ is a complex curve (which I will assume from now on), we can identify such a section with a section of the tensor square of $T^{*(1,0)}M$. Every quadratic differential $\omega$ defines the notion of positive tangent vectors $v\in T_pM$, namely, vectors satisfying $\omega(v,v)\ge 0$. If $\omega$ is holomorphic and does not vanish at $p$ then the set of positive tangent vectors in $T_pM$ is a real line $L_p\subset T_pM$. Therefore, one obtains a rank 1 smooth distribution (a line field) $L$ on the complement $M'$ to the set of zeroes of $\omega$ in $M$. (In other words, this is a real line subbundle in $TM'$.) This distribution $L$ is the field of "line elements" that Strebel is defining. The integrability condition of the Frobenius (integrability) theorem is trivially satisfied in this situation and, hence, $L$ is tangent to a foliation ${\mathcal H}$ on $M'$, called the horizontal foliation of $\omega$. Leaves of this foliation are the "integral curves" in Strebel's book.
Similarly, given an angle $\theta\in S^1$ one defines the distribution $L^{\theta}$ on $M'$ by the condition that a nonzero vector $v\in T_pM$ is in $L^{\theta}_p$ iff $$ arg(\omega(v,v))=\theta. $$ The line field $L^{\theta}$ is again tangent to a foliation on $M'$. Besides the horizontal foliation ($\theta=0$), one frequently uses the vertical foliation ($\theta=-1$). The horizontal and vertical foliations are orthogonal to each other with respect to the flat metric on $M'$ defined by $\omega$.