  We give a fairly elementary and simple proof that shows that the number of
incidences between $m$ points and $n$ lines in ${\mathbb R}^3$, so that no
plane contains more than $s$ lines, is $$ O\left(m^{1/2}n^{3/4}+
m^{2/3}n^{1/3}s^{1/3} + m + n\right) $$ (in the precise statement, the constant
of proportionality of the first and third terms depends, in a rather weak
manner, on the relation between $m$ and $n$).
  This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step
in their solution of Erd{\H o}s's distinct distances problem, is also a major
new result in incidence geometry, an area that has picked up considerable
momentum in the past six years. Its original proof uses fairly involved
machinery from algebraic and differential geometry, so it is highly desirable
to simplify the proof, in the interest of better understanding the geometric
structure of the problem, and providing new tools for tackling similar
problems. This has recently been undertaken by Guth~\cite{Gu14}. The present
paper presents a different and simpler derivation, with better bounds than
those in \cite{Gu14}, and without the restrictive assumptions made there. Our
result has a potential for applications to other incidence problems in higher
dimensions.
