How can I explain the enthalpy of a system?

Imagine you store a bunch of CDs in a plastic box. The structure of the box holds the CDs in place but does not specificaly exert forces on the CDs.

Next, take the CDs out of the box and throw them onto the table. There is no box to hold them neatly together so they just scatter across the table. To tidy them up a bit, you put an elastic band around the bunch of CDs. In stretching the elastic band around the bunch of CDs, you had to put some energy into the system. The CDs have to apply some pressure against the elastic band to stretch it and take up the space that the elastic band surrounds.
So in comparison to CDs in a box, the CDs now have to constantly push against the elastic band to prevent them being crushed by it.

OK, thats just an analogy, but you can use it to explain enthalpy.

If you imagine a gas inside a solid box, the energy is defined by internal energy U. The gas experiences no force from the sides of the box. But now release the gas from the box and expose it to the atmosphere. and the air exerts pressure from all sides, with air pressure P. So under this external pressure, in order to occupy volume V, the gas has to do work PV (pushing the air out of that specific volume).

For gas in an open system, we should consider two parts when thinking about its total energy:

(1) Internal energy U, and (2) the energy involved in overcoming pressure to occupy volume V, i.e. P.V.

The enthalpy H is the combined effect of these two factors, and is defined by H = U + PV.