HetNet stands for Heterogeneous Network, meaning 'a collection of all different networks'. Imagine you are making a list of all the network around you as you start from your home and arrives at a big office building.
Let's start from your home with WiFi. As you gets out of the house, you will have one or more LTE cell and 2G/3G Cells while you are driving to your office building. If your office building is very large, it is likely that their will be DAS (distributed antenna system), some pico/femoto cell and WiFi APs in the building. The collection of all of these networks can be called HetNet.
So What ?
I know I would have bunch of different network around me. So what ? Why do I have to care ?
Just a bunch of different network around you would not do any good for you.
However, what would you think if all of those networks are cooperating each other to give you a better/seamless/reliable communication experience as if you get connected to a single/never failing network ?
In short, the major motivation of HetNet is to boost network capability.
Now it sounds attractive, but
This is always the problem. As of now (Mar 2015), you wouldn't see the real implementation of HetNet as they promote. It would take at least a couple of more years before you start seeing even a portions of what business people is trying to promote. But I would recommend you to keep tracking this area. In my opinion, in 5G it would play an important role.
First, let's think of how we can increase network capability/capacity regardless of whether it is based on HetNet or any other means.
Just from the high level perspective, there are several idea we can think of as summarized in Nokia WhitePaper(Ref ) as listed below.
Item 1 and 3 are not HetNet based.. it is more like the way we have seen in the evolution of any cellular technology. In LTE advanced, we are already seeing the chipset and early engineering devices supporting 256 QAM as of now (Oct 2015) and has been testing for prettying long time already about the MIMO greather than 2 x 2. Now Downlink 4 x 4 has been a common test items and even Uplink MIMO (2 x 2) is being tested at UE level.
When it are talking about HetNet we normally refer to item 2 or 4, like WiFi Offload or LTE-U or Small Cells. WiFi Offload has been implemented/tested and now at the stage of early deployment in real network (As of Oct 2015) and LTE-U has been tested for several month.
 The HetNet Engine Room (H.E.R.): Bringing R.O.I. and Scalability to Ultra-Dense Networks - Nokia Whitepaper