A Hilbert space is the mathematical setting used to describe quantum states. Every possible state of a quantum system corresponds to a vector in this space, and the rules of quantum mechanics are expressed as operations on those vectors. For a single qubit the Hilbert space is simple enough to picture as the Bloch sphere, but for many qubits it quickly becomes far too large to visualise.
The key feature is how the space grows. Each qubit added to a system doubles the dimension of its Hilbert space, so n qubits require a space of two-to-the-power-n dimensions. Just 300 qubits describe a Hilbert space with more dimensions than there are atoms in the observable universe. Tracking a general state in this space is what makes quantum systems so expensive to simulate classically.
This exponential scaling is precisely what a quantum computer exploits. By using physical qubits to represent and manipulate states in this vast space directly, a quantum processor can perform computations that are intractable for classical machines. Harnessing it in practice depends on preserving the fragile quantum states involved, which is the central challenge of scaling real hardware.