What this slogan means, in its most common usage, is that there is no real difference between women’s political issues and their personal issues. It was used to argue that issues that people denigrated as personal issues that women should take care of on a personal level were really political issues that should be dealt with as a society.
In the late 1960s and early 1970s, many women’s issues were not taken very seriously by many people. When women wanted, for example, more help from men in terms of childcare and housework, it was seen as a personal issue. When women talked about the pressure that society put on them to be attractive and about the problems that caused with things like body image, the issues were dismissed as personal issues that women should deal with on their own.
“The personal is political” was used as a slogan to argue that all of these were truly political issues. It meant that all of these issues came about because of overall societal factors. When women wanted more help from men, it was an issue of society-wide attitudes about what constituted men’s work and women’s work. When they worried about body image, it was because of issues about how society portrays women and about what factors cause women to be valued or not valued.
Thus, the slogan was meant as a way of arguing that women’s issues should be taken more seriously on a political and social level.