Solenoidal vector fields can have the physical interpretation that if you draw any simple 3D solid, the net material going in and out of the solid is zero. That is, the amount of material inside the solid is conserved.
Irrotational vector fields can have the physical interpretation of "no whirlpools" or "no vertices".
In practical applications, the best ones are in electricity and magnetism. According to Maxwell's equations, magnetic fields are solenoidal, which leads to the "no magnetic monopoles" theory, although that's come to be challenged in recent years. If you have a solenoidal electric field, then you can use Gaussian surfaces to solve problems, because with a solenoidal electric field, there are no source charges.
I believe all this is correct. Someone else can certainly challenge me.