The sociological imagination refers to personal issues becoming public and social problems. For example one person being unemployed is devastating for the individual but 1000 people being unemployed causes areas of poverty and deprivation, which can then lead to an increase in crime, a rise in people claiming benefits etc etc.
If you think of it like when a pebble is thrown in a pond the ripples expand further and further outwards.
Durkhiem proved that sociology was a valuable subject by using the most personal act an individual can experience, ie. suicide and using social problems to explain the reasons for it. One study he used was to survey the rate of suicide in different countries such as Spain, France and Italy etc. He found that the rate of suicide in each country stayed near enough the same over each decade whereas the rates of suicide in each country where extremely different. He used religion to explain one reason for this for example in Italy where the dominant faith is catholic and suicide is considered to be a sin suicide is not a common factor. Whereas in countries where there is a predominantly protestant faith suicide is more common.
Sorry if this is not that clear, I have just been given an essay on the same thing and im finding it hard to relate it myself. This is what i have found on the subject up to now and these ideas can easily be found on the internet if you want to research them properly.