Numerically, it is a ratio between two numbers. Intuitively or physically, you can think of free space impedance as a limiting factor of the rate of change in electric/magnetic field. The following relation arises naturally from Maxwell equations:
$$\eta = \frac{E}{H}$$
where $\eta$ is the characteristic impedance of free space which is 377 Ohm, $E$ is the electric field of wave and $H$ is the magnetic field of the wave. You can compare that to Ohm’s law. If $\eta$ was set to zero, that means one of two things must be true:
Either the magnetic field ($H$) is infinity with finite electric field ($E$).
Or the electric field ($E$) is zero with non-zero magnetic field ($H$).
The first scenario describes a wave (none of the fields is zero) but it is wrong because it requires infinite $H$ which is nonphysical. Accordingly, having a non-zero value of $\eta$ limits the required magnetic field such that both $E$ and $H$ become non-zero and finite. The second scenario doesn't describe electromagnetic wave. It describes a magnetostaic situation.
The free space permittivity has a unit of $\mathrm{F/m}$ while free space permeability has a unit of $\mathrm{H/m}$. If you think of it you can model the propagation of EM wave in vacuum with an infinitely long circuit composed of inductance and capacitance. This circuit is drawn per unit of length that is why the units are given per meter. See the next figure

From a circuit perspective, the capacitor doesn’t allow sudden changes in the voltage, because that requires an infinite current (which is non-physical obviously) according to:
$$i_C = C \frac{\mathrm{d}v_C}{\mathrm{d}t}$$
In the same sense, inductance doesn't allow sudden changes in the current because that requires voltage (which is non-physical obviously) according to:
$$v_L = L \frac{\mathrm{d}i_L}{\mathrm{d}t}$$
The current and the voltage are equivalent to magnetic and electric fields respectively. Capacitance and inductance are equivalent to permittivity and permeability respectively. In such a circuit, the resonance frequency is equivalent to the speed of EM wave:
$$\omega = \frac{1}{\sqrt{LC}}\\c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$
Studying waves using circuit is a fully developed field; circuits used to study EM wave propagation not only in space but on any other media including transmission lines are called transmission lines circuits.
Hopefully that helped!