I'll try to cover his main points...
The various intelligence tests that exist show some positive correlation. The factor "g" is this degree of correlation. The best measure of g is Raven's Progressive Matrices (RPM), which is a test of visual reasoning. Intelligence tests are periodically revised and recalibrated. Flynn noticed that subjects were scoring higher on older versions. Scores on arithmetic, vocabulary, and general information showed small gains or even declines. The greatest gains were on g-loaded tests such as RPM.
Given the performance of current students in school subjects, such a rise in general intelligence doesn't seem plausible. Because of this, Flynn thinks the tests don't really measure intelligence.
Various theories have been put forward to account for the change. The gains were far too rapid to result from genetic changes. Another theory was that Americans became more familiar with test taking in general and that transferred to IQ tests, but the rise continued for decades after the spread of broader education. Also, the tests more closely related to school content were not the ones showing the rise.
Nutrition caused people to grow taller and have bigger heads, but there is very weak empirical support for a connection between nutrition and intelligence. Also, a purely biological basis for the change would indicate a rise in all aspects of intelligence, which clearly wasn't the case.
There is a well established effect of schooling on IQ, but populations of children before schooling have shown gains comparable to populations of adults after schooling. Also, groups of subjects at different levels of education showed about the same gains. Finally, schooling affects tests of content more than of reasoning, but that wasn't reflected in rising test scores.
Modern children increasingly have exposure to things explicitly designed to stimulate their intelligence, e.g. Sesame Street and preschool. Early childhood intervention programs have been shown to have no lasting effect on IQ. The most intensive environmental enrichment programs have been shown to have some lasting effect on IQ, but nowhere near comparable to the Flynn effect.
Neisser proposes that changes in the cultural environment have caused the rise. Modern industrial culture produces vastly more of complex visual media. He thinks our increasing practice in analyzing complex imagery has caused the gains in performance on visual reasoning tests.