The register's state is a single complex vector over the 8 basis states |q2 q1 q0⟩. Each gate button multiplies that vector by the real quantum-mechanical unitary matrix for H, X, Z, S or CNOT, applied only to the amplitude pairs that differ in the target qubit's bit — the same bookkeeping a real simulator backend uses, just written out by hand instead of as a dense matrix multiply.
The three "core" nodes on the right show each qubit's reduced density matrix (found by tracing out the other two qubits): the donut is its |0⟩/|1⟩ population, and the glow connecting two nodes is proportional to how far that qubit's own state has strayed from pure — i.e. how entangled it is with the others. A GHZ state (H on q0, then CNOT to q1 and q2) lights up all three links at once because no qubit has a well-defined state on its own anymore.
"Measure" collapses the vector: each basis state is picked with probability |amplitude|², matching the Born rule, and the bar chart below then shows a single spike at the outcome.