"What does it mean for a field to be excited?"
It means to be in something other than the quantum ground state. Modern physics conceives of the universe as being made of a handful of quantum fields. Space and time aren't just filled with quantum fields, they themselves are made of quantum fields. "Empty space" is simply a shorthand for what you have when the part of the quantum fields in question is in its ground state. These fields interact. It is this interaction we see when we observe "things happenning" in our World.
"Why does colliding particles at high energies produce these excitation?"
As I said, quantum fields interact. As one field's excitations drop towards ground state, there is a "coupling constant", an amplitude that another field's excitations rise to a "few particle" state. As the energies of collisions increase, the amplitude for a "swap" of excitations along these lines increases. So when we reach high enough energies, comparable to the Higgs particle's rest mass, the amplitude to observe a Higgs particle, i.e. the Higgs field in its once-removed-from-ground state, increases.