IT consultants perform a variety of functions, but the major items are implementation of new systems, architecture design, health checks, and knowledge transfer. Some firms offer longer term staffing solutions as well, but that typically isn't pure consulting.
Most large firm's IT staff are busy maintaining the systems they already have, and when they want to add a new complex technology they often don't have the time to become an expert on the new stuff and be able to design an architecture and implement it correctly the first time out.
Think of it like building a new house: The first time out of going to have some issues because of the learning curve. You want someone who has built that before. That's where consultants come in. A consulting firm would gather the requirements, design the architecture, build a project plan, provide experienced resources, and build the new system.The same consultants often do knowledge transfer as well. The clients IT staff may go to a couple of classes on the new software, but running it in the real world and understanding how it was architected takes a little one to one mentoring.
Over time, a firm may be called back in to do a health check and make sure things are still running smoothly and that the design is still appropriate to the business requirements and planned growth. They may also be called in to do an upgrade again as the software evolves.
A similar process can happen on the software development side. A client needs a custom software solution, and needs someone to architect the solution and lead the development effort.
The nice part of consulting is that every project is different, and you're always developing new skills. The travel is a bit brutal at times. But is is a rewarding and interesting career.