“Sometimes we just need that extra skilled resource“. 
When consulting I provide professional and expert advice, expertise and resource that you may not have within the business, or would not want to have full time.  This can can be a very cost effective way of engaging me for a service at a fixed price, where there is a clearly defined scope.
Sometimes I am also engaged because I am independent and can provide an objective view on dilemma’s and solutions, sometimes it is to take advantage of the fact that I am not susceptible to internal politics and/or sensitive situations.
Hiring consultants is normally cheaper than retaining the expertise in-house, so when you find yourself short in capacity for projects/positions consultants can backfill.
I can also take on the ‘dirty work’, i.e. take the decisions no one internally wants to make, or undertake tasks that would compromise internal staff such as staff restructuring projects, or objectively selecting staff for specific internal projects.