What does it mean by universe is expanding? I know that the universe is expanding because spacetime is expanding but again what does it mean?

Say some cosmic object spews forth a certain number of photons of a certain frequency, which make their way towards an observer.

The photon flux is a measure of distance: Assuming isotropy, it will decrease with $1/r^2$, constant across a growing spherical surface. The photon frequency is a measure of relative velocity, as evaluated by parallel transport through a curved spacetime.

In Friedmann universes specifically, spacetime can be sliced into spatial hypersurfaces of constant cosmological time, where the distance between 'stationary' points evolves according to a global scale factor. This distance can increase arbitrarily fast, and such 'recession velocities' can be greater than $c$. We can look beyond the Hubble sphere (the place where recession velocities hit the speed of light) without a problem. In contrast, relative velocities will always be smaller than $c$: When relative velocities approach the speed of light, redshift goes to infinity, and we cannot look beyond this cosmological event horizon.

Regarding your second point, note that you cannot just plug-in recession velocities into Doppler's formula and expect the correct answer: There's no distance parallelism in arbitrary spacetimes, and you'd have to integrate infinitesimal Doppler shifts along the photon trajectory. In Friedmann universes, this will yield a simple answer: Cosmological redshift between comoving emitters and observers ends up being anti-proportional to the scale factor at time of emission.

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Some historic remarks, prompted by the comments:

Einstein was looking for a relativistic theory of gravity, inventing general relativity. Friedmann then derived a certain class of maximally symmetric solutions to the Einstein equations, described by the FLRW metric and evolving accordig to the Friedmann equations. Einstein wasn't a fan, though, because he believed in a static univere (which necessitated the introduction of the cosmological constant).

Lemaître re-discovered these solutions, and, among other things, derived cosmological redshift. However, this calculation was omitted from the English translation of his paper, which meant few people where aware of it when Hubble derived the law that now bears his name heuristically: He (incorrectly) interpreted cosmological redshift special-relativistically, when the general-relaivistic interpretation had in fact already been proposed...