What led to the poverty increase during Reagan's years as president?
There are at least three different possible answers. We cannot know for sure which of them (or other possible answers) is the true answer. There is simply no way to know for a fact what causes something like this because there are too many factors that could be involved.
Liberals, who of course did not like Reagan, tend to blame the poverty on Reagan's cuts in government programs. They say that by cutting welfare and job training programs and such, he failed to help people get out of poverty.
Another possible explanation is that the economy became more globalized during the '70s and '80s. This led to rising poverty as relatively uneducated people (who used to be able to work for good wages in factories) lost their jobs to countries with cheaper wages.
Finally, you can say that it was all the liberal programs of the '60s and beyond that caused the poverty. Conservatives like to argue that the poor in the '80s were poor because government programs had encouraged them to be poor (by allowing them to survive even though they remained uneducated and without job skills).